# Notes on mssc-images of relatively compact metric spaces.

Abstract We introduce the notion of [sigma]-finite Ponomarev-system (f, M, X, {[P.sub.n]}) to give a consistent method to construct an mssc-mapping f with covering-properties onto a space X from some relatively compact metric space M. As applications, we systematically obtain characterizations on mssc-images of relatively compact metric spaces under certain covering-mappings, which sharpen results in [18, 26], and more.Keywords [sigma]-finite Ponomarev-system, relatively compact metric, sn-network, so-network,

cs-network, k-network, 2-sequence-covering, 1-sequence-covering, sequence-covering, compact-covering, mssc-mapping.

[section] 1. Introduction

An investigation of relations between spaces with countable networks and images of separable metric spaces is one of interesting questions on generalized metric spaces. Related to characterizing a space X having a certain countable network P by an image of a separable metric space M under some covering-mapping f, many results have been obtained. In the past, E. Michael [18] proved the following results.

Theorem 1.1. ([18], Proposition 10.2) The following are equivalent for a space X.

(1) X is a cosmic space.

(2) X is an image of a separable metric space.

Theorem 1.2. ([18], Theorem 11.4) The following are equivalent for a space X.

(1) X is an [N.sub.0]-space.

(2) X is a compact-covering image of a separable metric space.

Also, he posed the following question, on page 999.

Question 1.3. Find a better explanation for images of separable metric spaces? In [26], Y. Tanaka and Z. Li proved the following result.

Theorem 1.4. ([26], Corollary 8 (1)) The following are equivalent for a space X, where "subsequence-covering" can be replaced by "pseudo-sequence-covering".

(1) X has a countable cs*-network (resp., cs-network, sn-network).

(2) X is a subsequence-covering (resp., sequence-covering, 1-sequence-covering image of a separable metric space.

Recently, Y. Ge [8] sharpened Theorem 1.2 as follows.

Theorem 1.5. ([8], Theorem 12) The following are equivalent for a space X.

(1) X is an [N.sub.0]-space.

(2) X is a sequence-covering, compact-covering image of a separable metric space.

(3) X is a sequentially-quotient image of a separable metric space.

Taking these results into account, the following question naturally arises.

Question 1.6. When P is a countable network (cs-network, sn-network, so-network) for X; how nice can the mapping f and the metric domain M be taken to?

Around this question, the author of [2] has shown that, when P is a countable cs-network, f and M in Theorem 1.5 can be an mssc-mapping and a relatively compact metric space, respectively. More precisely, the following has been proved.

Theorem 1.7. ([2], Theorem 2.5] The following are equivalent for a space X.

(1) X is an [N.sub.0]-space.

(2) X is a sequence-covering, compact-covering mssc-image of a relatively compact metric space.

(3) X is a sequentially-quotient image of a separable metric space.

In the above result, f and M can not be any compact mapping and any compact metric space, respectively; see [2, Example 2.11 & Example 2.12]; and the key of the proof is to construct a sequence-covering, compact-covering mssc-mapping f from a relatively compact metric space M onto X.

By the above, we are interested in the following question.

Question 1.8. Find a consistent method to construct an mssc-mapping with covering-properties onto a space from some relatively compact metric space?

In this paper, we answer Question 1.8 by introducing the notion of o-finite Ponomarev-system (f, M, X, {[P.sub.n]}) to give a consistent method to construct an mssc-mapping f with covering-properties onto a space X from some relatively compact metric space M. As applications, we systematically obtain characterizations on mssc-images of relatively compact metric spaces under certain covering-mappings, which gives an answer for Question 1.6, particularly, for Question 1.3.

Throughout this paper, all spaces are regular and [T.sub.1], N denotes the set of all natural numbers, [omega] = N [union] {0}, and a convergent sequence includes its limit point. Let P be a family of subsets of X. Then [union]P, and [intersection]P denote the union [union]{P : P [member of] P}, and the intersection [intersection] {P: P [member of] P}, respectively. A sequence {[x.sub.n]: n [member of] [omega]} converging to [x.sub.0] is eventually in A [subset] X, if [x.sub.n] : n [greater than or equal to] [n.sub.0]} [union] {[x.sub.0]} [subset] A for some [n.sub.0] [member of] N, audit is frequently in A if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 1.9. Let P be a family of subsets of a space X.

(1) For each x [member of] X, P is a network at x in X, if x [member of] [intersection] P, and if x [member of] U with U open in X, then there exists P [member of] P such that S is eventually in P [subset] U.

(2) P is a cs-network for X [10] if, for every convergent sequence S converging to x [member of] U with U open in X, there exists P [member of] P such that S is eventually in P [subset] U.

(3) P is a cs*-network for X [5] if, for every convergent sequence S converging to x [member of] U with U open in X, there exists P [member of] P such that S is frequently in P [subset] U.

(4) P is a k-network of X [19] if, for every compact subset K [subset] U with U open in X, there exists a finite F [subset] P such that K [subset] U F [subset] U.

Definition 1.10. Let P = [union]{[P.sub.x] : x [member of] X} be a family of subsets of a space X satisfying that, for each x [member of] X, [P.sub.x] is a network at x in X, and if U, V [member of] [P.sub.x], then W [subset] U [intersection] V for some W [member of] [P.sub.x].

(1) P is a weak base for X [21], if G [subset] X such that for each x [member of] G, there exists P [member of] [P.sub.x] satisfying P [subset] G, then G is open in X.

(2) P is an so-network for X [14], if each member of [P.sub.x] is a sequential neighborhood of x in X.

(3) P is an so-network for X [12], if each member of [P.sub.x] is sequentially open in X.

(4) The above [P.sub.x] is respectively a weak base, an sn-network, and an so-network at x in X [12].

Definition 1.11. ([4]) Let P be a subset of a space X.

(1) P is a sequential neighborhood of x if, for every convergent sequence S converging to x in X, S is eventually in P.

(2) P is a sequentially open subset of X if, for every x [member of] P, P is a sequential neighborhood of x.

Definition 1.12. Let X be a space.

(1) X is relatively compact, if [bar.X] is compact.

(2) X is an No-space [18] (resp., so-second countable space [7], so-second countable space, g-second countable space [21], second countable space [3]), if X has a countable cs-network (resp., countable so-network, countable so-network, countable weak base, countable base.

(3) X is a sequential space [4], if every sequentially open subset of X is open.

Remark 1.13. ([15]) (1) For a space, weak base [??] so-network [??] cs-network.

(2) An so-network for a sequential space is a weak base.

Remark 1.14. (1) It is easy to see that "compact metric [??] relatively compact metric [??] separable metric", and these implications can not be reversed from Example 2.14 and Example 2.15.

(2) It is well-known that a space X is an No-space if and only if X has a countable k-network (cs*-network, see [23, Proposition C], for example.

Remark 1.15. Let f : X [right arrow] Y be a mapping.

(1) f is an mssc-mapping[13], if X is a subspace of the product space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of a family {[X.sub.n] : n [member of] N} of metric spaces, and for every y [member of] Y, there exists a sequence {[V.sub.y,n] : n [member of] N} of open neighborhoods of y in Y such that each [bar.[P.sub.n] ([f.sup.-1]([V.sub.y,n]n))] is a compact subset of [X.sub.n], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the projection.

(2) f is an 1-sequence-covering mapping [14] if, for every y [member of] Y, there exists [x.sub.y] [member of] [f.sup.-1](y) such that whenever {[y.sub.n] : n [member of] N} is a sequence converging to y in Y there exists a sequence {[x.sub.n] : n [member of] N} converging to [x.sub.y] in X with each [x.sub.n] [member of] [f.sup.-1](yn).

(3) f is a 2-sequence-covering mapping [14] if, for every y [member of] Y, x [member of] [f.sup.-1](y), and sequence {[y.sub.n] : n [member of] N} converging to y in Y, there exists a sequence {[x.sub.n] : n [member of] N} converging to x in X with each {[x.sub.n] [member of] [f.sup.-1](yn).

(4) f is an 1-sequentially quotient mapping [17] if, for every y [member of] Y, there exists [x.sub.y] [member of] [f.sup.-1](y) such that whenever {[y.sub.n] : n [member of] N} is a sequence converging to y in Y there exists a sequence {[x.sub.k] : k [member of] N} converging to [x.sub.y] in X with each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(5) f is a sequence-covering mapping [20] if, for every convergent sequence S in Y, there exists a convergent sequence L in X such that f (L) = S.

(6) f is a compact-covering mapping [18] if, for every compact subset K of Y, there exists a compact subset L of X such that f (L) = K.

(7) f is a pseudo-sequence-covering mapping [11], if for every convergent sequence S of Y, there exists a compact subset K of X such that f (K) = S.

(8) f is a sequentially-quotient mapping [1] if, for every convergent sequence S in Y, there exists a convergent sequence K in X such that f (K) is a subsequence of S.

For terms are not defined here, please refer to [3, 24].

[section] 2. Results

Definition 2.1. Let P = {[P.sub.n] : n [member of] N} be a countable network for a space X. Because X is [T.sub.1] and regular, we may assume that every member of P is closed. For each n [member of] N, let [P.sub.n] = {X} [union] {[P.sub.i] : i [less than or equal to] n} {[P.sub.[alpha]], : [alpha] [member of] [A.sub.n]}, where [A.sub.n] is a finite set, and let every [A.sub.n] be endowed with the discrete topology. Setting

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a network at some point [x.sub.a], [member of] X}.

Then M, which is a subspace of the product space [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is a metric space. Since X is [T.sub.1] and regular, [x.sub.a], is unique for each a [member of] M. We define f : M [right arrow] X by f(a) = [x.sub.a], for each a [member of] M. Then f is an mssc-mapping from a relatively compact metric space onto X by the following Theorem 2.3. The system (f, M, X, {[P.sub.n]}) is a o-finite Ponomarev-system.

Remark 2.2. For more details on Ponomarev-systems and images of metric spaces, see [9,16,25], for example.

Theorem 2.3. Let (f, M, X, {[P.sub.n]}) be the system in Definition 2.1. Then the following hold.

(1) M is a relatively compact metric space.

(2) f is an mssc-mapping.

Proof. (1) For every i [member of] N, since [A.sub.i], is finite, [A.sub.i] is compact metric. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is compact metric, and so [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is. It implies that M is relatively compact metric.

(2) It suffices to prove the following facts (a), (b), and (c).

(a) f is onto.

Let x [member of] X. Since P is a countable network for X, [(P).sub.x] = {P [member of] P : x [member of] P} is a countable network at x in X. We may assume that [(P).sub.x] = {[P.sub.x,j] : j [member of] N}, where [P.sub.x,j] [member of] [P.sub.i(j)] for some i(j) [member of] N satisfying i(j) < i(j+1). For each i [member of] N, taking [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some j [member of] N, and otherwise, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a network at x in X. Setting a = ([[alpha].sub.i]), we get a [member of] M and f(a) = x.

(b) f is continuous.

Let x = f (a) [member of] U with U open in X and a [member of] M. Setting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a network at x in X. Then there exists n [member of] N such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Setting [M.sub.a] = {b - ([[beta].sub.i]) [member of] M : [[beta].sub.n] = [[alpha].sub.n]}, then [M.sub.a] is an open neighborhood of a in M. For each b [member of] [M.sub.a], we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It implies that f ([M.sub.a]) [subset] U. (c) f is an mssc-mapping.

Let x [member of] X. For each i [member of] N, taking [V.sub.x,i] = X. Then {[V.sub.x,i] : i [member of] N} is a sequence of open neighborhoods of x in X. Since [A.sub.i], is finite, [A.sub.i] is compact. Then [bar.[p.sub.i](sup.f-1]([V.sub.x,i]))] = [p.sub.i]([f.sup.-1] (X)) [subset] [A.sub.i] is compact. It implies that f is an mssc-mapping.

Remark 2.4. In the view of (a) in the proof (1) [??] (2) of [2, Theorem 2.5], f is onto by choosing a = ([[alpha].sub.i]) satisfying f(a) = x, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Unfortunately, this argument is not true. In fact, we may pick [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] does not form a network at x in X. It is a contradiction. The wrong argument is corrected as in the proof (2).(a) of Theorem 2.3.

Theorem 2.5. Let (f, M, X, {P.}) be a [sigma]-finite Ponomarev-system. Then the following hold.

(1) If P is a cs-network for X, then f is sequence-covering.

(2) If P is a k-network for X, then f is a compact-covering.

(3) If P is a cs*-network for X, then f is pseudo-sequence-covering.

(4) If P is an sn-network for X, then f is 1-sequence-covering.

(5) If P is an so-network for X, then f is 2-sequence-covering.

Proof. (1) Let S = {[x.sub.m] : m [member of] [omega]} be a convergent sequence converging to [x.sub.0] in X. Suppose that U is an open neighborhood of S in X. A family A of subsets of X has property Cs (S, U) if.

(i) A is finite.

(ii) for each P [member of] A, 0 [not equal to] P [intersection] S [subset] P [subset] U.

(iii) for each [x.sub.m] [member of] S, there exists unique [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(iv) if [x.sub.0] [member of] P [member of] A, then S - P is finite.

For each i [member of] N, since A = {X} C [P.sub.i] has property cs(S, X) and [P.sub.i] is finite, we can assume that

{A [subset] [P.sub.i] : A has property cs(S, X)} = {[A.sub.i(j)] : j - [n.sub.i-1] + 1, ..., [n.sub.i]},

where [n.sub.0] = 0. By this notation, for each j [member of] N, there is unique i(j) [member of] N such that [A.sub.i(j)]) has property cs(S, X). Then for each j [member of] N, we can put [A.sub.i(j)] {[P.sub.[alpha]], : [alpha] [member of] [E.sub.j]}, where [E.sub.j] is a finite subset of [A.sub.j].

For each j [member of] N, m [member of] [omega] and [x.sub.m] [member of] S, it follows from (iii) that there is unique [[alpha].sub.jm]) [member of] [E.sub.j] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We shall prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a network at [x.sub.m] in X. In fact, let [x.sub.m] E U with U open in X, we consider two following cases.

(a) If m = 0, then S is eventually in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. For each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], let x [member of] [P.sub.x] [subset] X - (S- {x}) for some [P.sub.x] [member of] P. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] has property cs(S, X). Since G is finite, G [subset] [P.sub.i] for some i [member of] N. It implies that G = [A.sub.i(j)] for some i(j) [member of] N with j [member of] {[n.sub.i-1] + 1, ..., [n.sub.i]}. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By the above, for each m [member of] [omega] we get [a.sub.m], ([[alpha].sub.j,m]) [member of] M satisfying f ([a.sub.m]) = [x.sub.m]. For each j [member of] N, since families H and G are finite, there exists [m.sub.j] [member of] N such that ([[alpha].sub.j,m]), = [[alpha].sub.j,0] if m [greater than or equal to] [m.sub.j]. Hence the sequence {[[alpha].sub.j,m] : m [member of] N} converges to [[alpha].sub.j,0] in [A.sub.j]. Thus, the sequence {[a.sub.m] : m [member of] N} converges to [a.sub.0] in M. Setting L = {[a.sub.m] : m [member of] [omega]}, then L is a convergent sequence in M and f (L) = S. This shows that f is sequence-covering.

(2) Let K be a compact subset of X. Suppose that V is an open neighborhood of K in X. A family B of subsets of X has property k(K, V) if.

(i) B is finite.

(ii) P [intersection] K [not equal to] 0 for each P [member of] B.

(iii) K [subset] [union] B [subset] V.

For each i [member of] N, since B = {X} C [P.sub.i] has property k(K, X) and [P.sub.i] is finite, we can assume that

{B [subset] [P.sub.i] : B has property k (K, X)} {[B.sub.i(j)]: j = [n.sub.i-1] + 1, ..., [n.sub.i]}, where [n.sub.0] = 0. By this notation, for each j [member of] N, there is unique [B.sub.i(j)] [member of] N such that [B.sub.i(j)] has property k (K, X). Then for each j [member of] N, we can put [B.sub.i(j)] = {[P.sub.[alpha]] : [alpha] [member of] [F.sub.j]}, where [F.sub.j] is a finite subset of [A.sub.j].

Setting [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we shall prove that L is a compact subset of M satisfying f (L) = K, hence f is compact-covering, by the following facts (a), (b), and (c).

(a) L is compact.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This is a contradiction of the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(b) L [subset] M and f(L) [subset] K.

Let a - ([[alpha].sub.i]) [member of] L, then a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Taking [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a network at x in X, then a [member of] M and f(a) = x, hence L [subset] M and f (L) [subset] K. So we only need to prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a network at x in X. Let V be an open neighborhood of x in X. There exist an open subset W of K such that x [member of] W, and compact subsets [[bar.W].sup.K] and [[bar.W].sup.K] such that [[bar.W].sup.K] [subset] V and K - W [subset] X - {x}, where [[bar.W].sup.K] is the closure of W in K. Since P is a k-network for X, there exist finite families [F.sub.1] [subset] P and [F.sub.2] [subset] P such that [[bar.W].sup.K] C [union] [F.sub.1] [member of] V and K - W [subset] [union] [F.sub.2] [subset] X - {x}. We may assume that P [intersection] K [not equal to] 0 for each P [member of] [F.sub.1] [union] [F.sub.2]. Setting F = [F.sub.1] [union] [F.sub.2], then F has property k(K, X). It implies that F = [B.sub.i(j)] for some i(j) [member of] N with j [member of] {[n.sub.i-1] + 1, ..., [n.sub.i]}. Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This prove that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a network at x in X.

(c) K [union] f (L).

Let x [member of] K. For each i [member of] N, there exists [[alpha].sub.i] [member of] [F.sub.i] such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Setting a = ([[alpha].sub.i]), then a [member of] L. Furthermore, f (a) = x as in the proof of (b). So K [subset] f (L).

(3) Let S be a convergent sequence of X. By using [26, Lemma 3] and as in the proof of (2), here S plays the role of the compact subset K, there exists a compact subset L of M such that f (L) = S. Then, f is pseudo-sequence-covering.

(4) Let P = [union]{[P.sub.x] : x [member of] X}, where each [P.sub.x] is an sn-network at x in X. Let x [member of] X. We may assume that [P.sub.x] = {[P.sub.x,j] : j [member of] N}, where [P.sub.x,j] [member of] [P.sub.i(j)] with some i(j) [member of] N satisfying i(j) < i(j+1). For each i [member of] N, take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if i = i(j) for some j [member of] N, and otherwise, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a network at x in X. Setting [a.sub.x] = ([[alpha].sub.i]), then a [member of] M and [a.sub.x] [member of] [f.sup.-1](x). For each n [member of] N, setting [B.sub.n] = {b = ([[beta].sub.i]) [member of] M : [[beta].sub.i] = [[alpha].sub.i] if i [less than or equal to] n}. Then {[B.sub.n] : n [member of] N} is a decreasing neighborhood base at [a.sub.x] in M. For each n [member of] N, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In fact, for each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Conversely, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] then x = f (b) for some b = ([[beta].sub.i]) [member of] M. Setting c = ([[gamma].sub.i]), where [[gamma].sub.i] = [[alpha].sub.i] if i [less than or equal to] n, and [[gamma].sub.i] = [[beta].sub.i-n] if i > n. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a network at x in X. For each i [greater than or equal to] n, since [[beta].sub.i-n] [member of] [A.sub.i-n] [subset] [A.sub.i], [[gamma].sub.i] [member of] [A.sub.i]. Then c [member of] [B.sub.n] and f(c) = x. It implies that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Suppose that the sequence {[x.sub.j] : j [member of] N} converges to x in X. For each i [member of] N, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a sequential neighborhood of x in X, there exists [j.sub.i] [member of] N such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each j [greater than or equal to] [j.sub.i]. For each n [member of] N, setting j(n) = max{[j.sub.i] : i [less than or equal to] n}, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each j [greater than or equal to] j(n).

It implies that [f.sup.-1]([x.sub.j]) [intersection] [B.sub.n] [not equal to] 0 for each j [greater than or equal to] j(n). We may assume that 1 < j(n) < j(n + 1). For each j [member of] N, taking

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Then the sequence {[a.sub.j] : j [member of] N} converges to [a.sub.x] in M. In fact, let U be a neighborhood of [a.sub.x], then there exists n [member of] N such that [a.sub.x] [member of] [B.sub.n] [subset] U. Obviously, [a.sub.j] [member of] [B.sub.n] [subset] U for each j [greater than or equal to] j(n), so {[a.sub.j] : j [member of] N} converges to [a.sub.x]. Note that f ([a.sub.j]) = [x.sub.j] for every j [member of] N. It implies that f is 1-sequence-covering.

(5) Let x [member of] X and [a.sub.x] [member of] [f.sup.-1](x). Setting [a.sub.x] = ([[alpha].sub.i]), then [[alpha].sub.i] [member of] [A.sub.i] for each i [member of] N, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] forms a network at x in X. For each n [member of] N, setting [B.sub.n] = {b = ([[beta].sub.i]) [member of] M : [[beta].sub.i] = [[alpha].sub.i] if i [less than or equal to] n}. Then {[B.sub.n] : n [member of] N} is a decreasing neighborhood base at [a.sub.x] in M, and for each n [member of] N, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as in the proof of (4).

Suppose that sequence {[x.sub.j] : j [member of] N} converges to x in X. For each i [member of] N, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is sequentially open, there exists [j.sub.i] [member of] N such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for each j [greater than or equal to] [j.sub.i]. As in the proof (4), we get a sequence {[a.sub.j] : j [member of] N} converges to [a.sub.x] in M such that f ([a.sub.j]) = [x.sub.j] for each j [member of] N. It implies that f is 2-sequence-covering.

By using Theorem 2.3 and Theorem 2.5 above, we obtain characterizations of mssc-images of relatively compact metric spaces under certain covering-mappings, which sharpen results in [18,26] and more. Firstly, we sharpen Theorem 1.1 as following.

Corollary 2.6. The following are equivalent for a space X.

(1) X is a cosmic space.

(2) X is an mssc-image of a relatively compact metric space.

(3) X is an image of a separable metric space.

Proof. We only need to prove (1) [??] (2), other implications are routine. Since X is a cosmic space, X has a countable network P. Then the o-finite Ponomarev-system (f, M, X, {[P.sub.n]}) exists. Therefore, f is an mssc-mapping and M is relatively compact metric by Theorem 2.3. It implies that X is an mssc-image of a relatively compact metric space.

Remark 2.7. Corollary 2.6 is an answer of (question 1.3.

Next, we extend partly Theorem 1.4 as follows.

Corollary 2.8. The following are equivalent for a space X.

(1) X is an sn-second countable space.

(2) X is an 1-sequence-covering, compact-covering mssc-image of a relatively compact metric space.

(3) X is an 1-sequentially-quotient image of a separable metric space.

Proof. (1) [??] (2). Since X is an sn-second countable space, X has a countable closed sn-network [P.sub.1] and a countable closed k-network [P.sub.2]. Set P = [P.sub.1] [union] [P.sub.2], then P is a countable sn-network and k-network for X. It follows from Definition 2.1 that the [sigma]-finite Ponomarev-system (f, M, X, {[P.sub.n]}) exists. Therefore, f is an 1-sequence-covering, compact-covering mssc-mapping from a relatively compact metric space M onto X by Theorem 2.3 and Theorem 2.5.

(2) [??] (3). It is obvious.

(3) [??] (1). Let f : M [right arrow] X be an 1-sequentially-quotient mapping from a separable metric space M onto X. Since M is separable metric, M has a countable base B. We may assume that B is closed under finite intersections. For each x [member of] X, there exists [a.sub.x] [member of] [f.sup.-1](x) such that whenever {[x.sub.n] : n [member of] N} is a sequence converging to x in X there exists a sequence {[a.sub.k] : k [member of] N} converging to [a.sub.x] in M with each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [B.sub.x] = {B [member of] B : [a.sub.x] [member of] B} and [P.sub.x] = f ([B.sub.x]). We shall prove that P = [union] {[P.sub.x] : x [member of] X} is a countable sn-network for X. Since B is countable, P is countable. It suffices to prove the following facts (a), (b), and (c) for every X [member of] X.

(a) [P.sub.x] is a network at x in X.

It is straightforward because Lax is a neighborhood base at [a.sub.x] in M.

(b) If [P.sub.1], [P.sub.2] [member of] [P.sub.x], then P [subset] [P.sub.1] [intersection] [P.sub.2] for some P [member of] [P.sub.x].

Let [P.sub.1] = f ([B.sub.1]), [P.sub.2] = f ([B.sub.2]) with [B.sub.1], [B.sub.2] [member of] [B.sub.x]. Then [B.sub.1] [intersection] [B.sub.2] [member of] [B.sub.x]. Setting P = f([B.sub.1] [intersection] [B.sub.2]), we get P [member of] [P.sub.x] and P [subset] [P.sub.1] [intersection] [P.sub.2].

(c) Each P [member of] [P.sub.x] is a sequential neighborhood of x.

Let P = f (B) with B [member of] [B.sub.x], and let {[x.sub.n] : n [member of] N} be a sequence converging to x in X. For each subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of {[x.sub.n] : n [member of] N}, since f is 1-sequentially-quotient, there exists a sequence {[a.sub.k] : k [member of] N} converging to [a.sub.x] in M with each [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Since {[a.sub.k] : k [member of] N} [union] {[a.sub.x]} is eventually in B, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is frequently in P. Then {[x.sub.n] : n [member of] N} [union] {x} is frequently in P. It follows from [6, Remark 1.4] that P is a sequential neighborhood of x.

Corollary 2.9. The following are equivalent for a space X.

(1) X is a g-second countable space.

(2) X is an 1-sequence-covering, compact-covering, quotient mssc-image of a relatively compact metric space.

(3) X is an 1-sequentially-quotient, quotient image of a separable metric space.

Proof. (1) [??] (2). It follows from Corollary 2.8 that X is an 1-sequence-covering mssc-image of a relatively compact metric space under some mapping f. Since f is an 1-sequence-covering mapping onto a g-second countable space, f is a quotient mapping by [15, Lemma 3.5]. It implies that X is an 1-sequence-covering, quotient mssc-image of a relatively compact metric space.

(2) [??] (3). It is obvious.

(3) [??] (1). Since X is a quotient image of a separable metric space, X is a sequential space by [4, Proposition 1.2]. Then X is a sequential space having a countable sn-network P by Corollary 2.8. It follows from Lemma 1.13 that P is a weak base for X. Then X is a g-second countable space.

In the following, we prove a result on spaces having countable so-networks.

Corollary 2.10. The following are equivalent for a space X.

(1) X is an so-second countable space.

(2) X is a 2-sequence-covering, compact-covering mssc-image of a relatively compact metric space.

(3) X is a 2-sequence-covering image of a separable metric space.

Proof. (1) [??] (2). Since X is an so-second countable space, X has a countable closed so-network [P.sub.1] and a countable closed k-network [P.sub.2]. Set P = [P.sub.1] [union] [P.sub.2], then P is a countable so-network and k-network for X. It follows from Definition 2.1 that the [sigma]-finite Ponomarev-system (f, M, X, {[P.sub.n]}) exists. Therefore, f is a 2-sequence-covering, compact-covering mssc-mapping from a relatively compact metric space M onto X by Theorem 2.3 and Theorem 2.5.

(2) [??] (3). It is obvious.

(3) [??] (1). Let f : M [right arrow] X be a 2-sequence-covering mapping from a separable metric space M onto X. Since M is separable metric, M has a countable base B. For each x [member of] X, let [B.sub.x] = {B [member of] B : [f.sup.-1](x) [intersection] B [not equal to] 0}, and [P.sub.x] be the family of finite intersections of members of f ([B.sub.x]). We shall prove that P = [union] {[P.sub.x] : x [member of] X} is a countable so-network for X. Since B is countable, P is countable. It suffices to prove the following facts (a), (b), and (c) for every X [member of] X.

(a) [P.sub.x] is a network at x in X.

It is straightforward because Lax is a base of [f.sup.-1](x) in M.

(b) If [P.sub.1], [P.sub.2] [member of] [P.sub.x], then P [subset] [P.sub.1] [intersection] [P.sub.2] for some P [member of] [P.sub.x].

It is straightforward by choosing P = [P.sub.1] [intersection] [P.sub.2].

(c) Each P [member of] [P.sub.x] is sequentially open.

For each y [member of] f (B) with B [member of] [B.sub.x], we get [f.sup.-1](y) [intersection] B [not equal to] 0. Let {[y.sub.n] : n [member of] N} be a sequence converging to y in X. Then there exists a sequence {[a.sub.n] : n [member of] N} converging to a [member of] [sup.f.-1](y) [intersection] B such that f ([a.sub.n]) = [y.sub.n] for each n [member of] N. Since B is open, {[a.sub.n] : n [member of] N} [union] {a} is eventually in B. It implies that {[y.sub.n] : n [member of] N} [union] {y} is eventually in f(B). Hence f(B) is a sequential neighborhood of y in X. Therefore, f(B) is sequentially open. Since each P [member of] [P.sub.x] is some intersection of finitely many members of f ([B.sub.x]), P is sequentially open.

Corollary 2.11. The following are equivalent for a space X.

(1) X is a second countable space.

(2) X is a 2-sequence-covering, compact-covering, quotient mssc-image of a relatively compact metric space.

(3) X is a 2-sequence-covering, quotient image of a separable metric space.

Proof. (1) [??] (2). It follows from Corollary 2.10 that X is a 2-sequence-covering mssc-image of a relatively compact metric space under some mapping f. Since f is a 2-sequence-covering mapping onto a second countable space, f is a quotient mapping by [15, Lemma 3.5]. It implies that X is a 2-sequence-covering, quotient mssc-image of a relatively compact metric space.

(2) [??] (3). It is obvious.

(3) [??] (1). Since X is a quotient image of a separable metric space, X is a sequential space by [4, Proposition 1.2]. Then X is a sequential space having a countable so-network P by Corollary 2.10. For each P [member of] P, since P is sequentially open and X is sequential, P is open. Then P is a countable base for X, i.e., X is a second countable space.

Remark 2.12. By using Theorem 2.3 and Theorem 2.5, we also get the proof of Theorem 1.7 again, which was presented in [2].

From the above results, we obtain preservations of certain spaces under covering-mappings as follows.

Corollary 2.13. Let f : X [right arrow] Y be a mapping. Then the following hold.

(1) If X is an sn-second countable space and f is 1-sequentially-quotient, then Y is an sn-second countable space.

(2) If X is a g-second countable space and f is 1-sequentially-quotient, quotient, then Y is a g-second countable space.

(3) If X is an so-second countable space and f is 2-sequence-covering, then Y is an so-second countable space.

(4) If X is a second countable space and f is 2-sequence-covering, quotient, then Y is a second countable space.

Proof. We only need to prove (1), by similar arguments, we get the other. Since X is an sn-second countable space, X is an 1-sequentially-quotient image of a separable metric space under some mapping g by Corollary 2.8. It is easy to see that f o g is also an 1-sequentially quotient mapping. Then Y is an 1-sequentially-quotient image of a separable metric space under the mapping f o g. Therefore, Y is an sn-second countable space by Corollary 2.8.

Finally, we give examples to illustrate the above results.

Example 2.14. ([2], Example 2.8) A relatively compact metric space is not compact.

Example 2.15. ([2], Example 2.9) A separable metric space is not relatively compact.

Example 2.16. A 2-sequence-covering, compact-covering mapping from a separable metric

space is not an mssc-mapping.

Proof. Let f be the mapping in [2, Example 2.10]. Then f is also 2-sequence-covering. This complete the proof.

Example 2.17. A second countable space is not any image of a compact metric space. It implies that "relatively compact metric" in the above results can not be replaced by "compact metric".

Proof. Let 18 be the set of all real numbers endowed with the usual topology. Then R is a second countable space. Since 18 is not compact, R is not any image of a compact metric space. Example 2.18. A g-second countable space is not any 1-sequence-covering, compact-covering compact image of a separable metric space. It implies that "mssc-image" in Corollary 2.8 can not be replaced by "compact image".

Proof. See [22, Example 2.14(3)].

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Nguyen Van Dung

Department of Mathematics, Pedagogical University of Dongthap, Caolanh City, Dongthap Province, Vietnam

nguyendungtc@yahoo.com, nvdung@pud.edu.vn

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Author: | Van Dung, Nguyen |
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Publication: | Scientia Magna |

Article Type: | Report |

Geographic Code: | 9VIET |

Date: | Sep 1, 2008 |

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