# Separation axioms in fibrewise proximity spaces.

Introduction

The history of fibrewise topology goes back to Riemann who introduced the concept. The modern development of the subject has its origin in the work of Hurewicz on fibre spaces and of Whitney of fibre bundles.

Fibrewise uniform spaces were studied by Niefield where both base space and the total space are required to be uniform and the projection is required to be uniformly continuous.

In fibrewise topology we work over a topological base space B. When B is a point space, the theory reduces to that of ordinary topology. For results and definitions related to the theory of fibrewise topological space, we refer [1].

Preliminaries and Definitions

The present section contains some of the basic notions required for proximity spaces. Definition needed for the forthcoming sections are defined in this section.

2.1 Definition: A binary relation 8 on the power set of X is called (Efremovic) proximity on X iff [delta] satisfies the following axioms-

i) A[delta]B [??] B[delta]A;

ii) (A [union] B)[delta]C iff A[delta]C or B[delta]C;

iii) A[delta]B [??] A [not equal to] [phi], B [not equal to] [phi];

iv) A[delta]B [??] there exists a subset E such that A[delta]E and (X -- E)[delta]B;

v) A [intersection] B [not equal to] [phi] [??] A[delta]B.

The pair (X, [delta]) is called a proximity space. If in addition, [delta] satisfies.

vi) x[delta]y [??] x = y,

then (X, [delta]) is called s separated proximity space.

From now onwards by a proximity space we shall mean a separated proximity space unless otherwise stated.

2.2 Remark: Proximity [delta] on X induces a topology [tau] = [tau]([delta]) on X if for a subset of X, its closure is defined to be the set.

[bar.A] = {x: x[delta]A}

Note that this topology is always completely regular.

Conversely, if (X, [tau]) is any completely regular space, then there exists a proximity [delta] on X such that [tau] = [tau]([delta])

2.3 Definitions: A subset B of a proximity space (X, [delta]) is a [delta]-neighbourhood of A, denoted by A << B iff A[delta]X-B.

Following results are stated without proof:

2.4 Let (X, [delta]) be a proximity space. Then axiom (iv) in definition 2.1 is equivalent to A[delta]B implies there exists subsets C and D such that A[delta](X -- C),(X -- D)[delta]B and C[delta]D

2.5 In a proximity space (X, [delta]), if A[delta]B, A [subset] B, and B [subset] C, then C[delta]D.

2.6 Definition: A function [phi]: X [right arrow] Y is said to be p-open if for each A, B [subset or equal to] [X.sub.w], where W [subset or equal to] B,

A << B [??] [phi](A) << [phi](B).

2.7 Definition: A function [phi]: X [right arrow] Y is said to be p-closed if for A, B [subset] [X.sub.w], where W [subset or equal to] B, A[delta]B [??] [phi](A)[delta][phi](B)

2.8 Definition: If (X, [delta]) is a proximity space and Y [subset] X, then subsets of Y are also subsets of X. For subsets A and B of Y, we define.

A[[delta].sub.Y]B iff A[delta]B

Fibrewise Proximity Space

In the present section separation axioms in the fibrewise proximity spaces have been introduced.

3.1 Definition: A fibrewise set over a set B, called the base set, consists of a set X together with a mapp: X [right arrow] B, called the projection. For a point b [member of] B, [p.sub.-1](b), denoted by Xb, is called the fibre of b.

Also for W [subset or equal to] B, [p.sub.-1](w), is denoted by [x.sub.w].

3.2 Definition: Let B be a proximity space. By a fibrewise proximity on a fibrewise set X over B we mean any proximity on X which makes the projection p-continuous.

3.3 Definition: A fibrewise proximity space X over B is a fibrewise set over B with a fibrewise proximity.

Result is stated without proof:

3.4 Let [phi]: X [right arrow] Y be an p-open and p-closed fibrewise onto, where, X and Y are fibrewise proximity spaces over B. Let [alpha]: X [right arrow] R be a p-map, which is bounded above in the sense that a is bounded above on the each fibre on X i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then [beta]: Y [right arrow] R is p-map, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

3.5 Definition: Let X and Y be fibrewise proximity spaces over the base space B. A map [phi]: X [right arrow] Y is said to be fibrewise if [phi](Xb) [subset or equal to] [Y.sub.b].

3.6 Definition: A map [phi]: X [right arrow] Y is said to fibrewise p-continuous if for A [subset or equal to] [X.sub.w] where W [subset or equal to] B and for any proximal neighbourhood K of [phi](A), its inverse image, [phi]-1(K) is a proximal neighbourhood of A i.e.

[phi](A) << K [??] A << [[phi].sup.-1](K).

When A = {x}, the above condition reduces to

[phi](x) << K [phi] x << [[phi].sup.-1](K)

Note that a fibrewise p-continuous map is a fibrewise continuous map with respect to the induced topologies.

Separation Axioms

In the present section separation axioms in fibrewise proximity spaces are defined and some results have been proved.

4.1. Definition: Let X be a FPS. Then X is called a fibrewise T0-proximity space if for each fibre Xb, b [member of] B is T0. In other words, X is T0, if whenever x1, x2 [member of] Xb, b [member of] B,, and x1 [not equal] x2 there exists x1 << Gx1, x2 [not member of] Gx2 or x2[delta]X - Gx2.

4.2 Theorem: -T0-ness is hereditary in FPS.

Proof: Let (Y, [[delta].sub.Y]) be a subspace of a FPS X. Let y1, y2 [member of] Yb, b [member of] B such that y1 [not equal to] y2. Since X is fibrewise T0, there exists a proximal neighbourhood say H of y1 in X such that y2 [not member of] H, i.e. y1 << H and y2 [not member of] H or y2[delta]X-H.

This gives y1 << H [intersection] Y and y2. [not member of] H [intersection] Y. Then there exists a proximal neighbourhood H [intersection] Y of y1 in Y such that y2 [not member of] H [intersection] Y. Hence H is fibrewise TO.

4.3 Definition: Let X be a FPS. Then X is said to be fibrewise T1 if it's each fibre is T1, if for [x.sub.1], [x.sub.2] [[member of].sub.b], b [member of] B there exists a disjoint proximal neighbourhood V1 and V2 of [x.sub.1] and [x.sub.2] in [X.sub.w] such that [x.sub.1] << [X.sub.w] [intersection] V1 but [x.sub.2] [not member of] [X.sub.w] [intersection] [V.sub.1] and vice versa.

4.4 Definition: The fibrewise proximity space X over B is fibrewise Ro if for each point x [not member of] [X.sub.b], b [not member of] B, and for each proximal neighbourhood V >> x in X, there exists a proximal neighbourhood W >> b in B such that [X.sub.w] [intersection] ([bar.x]) << V.

4.5 Theorem: Let [phi]: X [right arrow] Y be a fibrewise p-map, where X and Y are fibrewise proximity spaces over the same base B. If Y is fibrewise Ro-proximity space. Then so is X.

Proof: Let x [member of] [X.sub.b], b [member of] B and let V be a proximal neighbourhood of x in S.

Since [phi] is onto. Choose V = [[phi].sup.-1]V', where V' >> [phi](x) in Y.

Since Y is fibrewise Ro, there exists a proximal neighbourhood W >> b in B such that [Y.sub.w] [intersection] (y) << V'. Then [[phi].sup.-1] ([Y.sub.w] [intersection] ([bar.y])) << [[phi].sup.-1]V' = V.

[??][X.sub.w] [intersection] ([bar.x]) << V

Thus, for any x [member of] [X.sub.b], b [member of] B and V >> x in X, there exists a proximal neighbourhood W >> b in B such that [X.sub.w][intersection] ([bar.x]) << V. So, X is fibrewise Ro-proximity space.

4.6 Theorem: Let [phi]: X [right arrow] Y be a p-closed and p-continuous fibrewise onto map, where X and Y are fibrewise proximity spaces over B. If X is fibrewise Ro-proximitiy space, then so is Y.

Proof: Let y [member of] [Y.sub.b], b [member of] B and let V >> y in Y. As [phi] is onto. Choose x [member of] [[phi].sup.-1](y).

Since [phi] is p-continuous map. Therefore, x << [[phi].sup.-1](V) in X.

By hypothesis, X is fibrewise Ro-proximity space, there exists a proximal neighbourhood W >> b in B such that

[X.sub.w] [intersection] ([bar.x]) << [[phi].sup.-1](V)

[??] [Y.sub.w] [intersection] [phi]([bar.x]) << [phi]([[phi].sup.-1](V)) = V

[??] [Y.sub.w] [intersection] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] << V (since [phi] is closed).

So Y is fibrewise Ro-proximity space.

4.7 Definition: Let X be a FPS. Then X is a fibrewise Hausdroff (separated), if whenever [x.sub.1], [x.sub.2] [member of] [X.sub.b], b [member of] B such that [x.sub.1] [not equal to] [x.sub.2]. Then there exists a proximal neighbourhood V1 and V2 of [x.sub.1] and [x.sub.2] in X.

4.8 Theorem: Let [phi]: X [right arrow] Y be p-continuous, fibrewise injective map, where X and Y are fibrewise proximity spaces over B. If Y is fibrewise separated. Then so in X.

Proof: Let [x.sub.1], [x.sub.2] [member of] [X.sub.b], b [member of] B and [x.sub.1] [not equal to] [x.sub.2]. Since [phi] is injective. Therefore, [phi]([x.sub.1]), [phi]([x.sub.2]) [member of] [Y.sub.b] and [phi]([x.sub.1]) [not equal to] [phi]([x.sub.2]). Since Y is fibrewise separated, so. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then there exists subsets A and B of Y such that [phi]([x.sub.1]) << A, [phi]([X.sub.2]) << B and A[phi]B. Since 0 is p-continuous, it follows that [x.sub.1] << [[phi].sup.-1](A), [x.sub.2] << [[phi].sup.-1](B) and [[phi].sup.-1](A) [intersection] [[phi].sup.-1](B) = [phi]. Consequently, X is fibrewise separated.

4.9 Definition: The FPS X over B is fibrewise functionally separated (FFS), if whenever [x.sub.1], [x.sub.2] [member of] [X.sub.b], b [member of] B and [x.sub.1] [not equal to] [x.sub.2] there exists a proximal neighbourhood W >> b and a p-continuous map [alpha]: [X.sub.w] [right arrow] I such that

[alpha]([x.sub.1]) = 0 and [alpha]([x.sub.2]) = 1.

4.10 Theorem: Subspaces of fibrewise functionally separated proximity space are fibrewise functionally separated.

Proof: Let X be a FFSPS over B and Y be a fibrewise proximal subspace of X. Then Y is also a FPS over the base space B. Let [y.sub.1], [y.sub.2] [member of] [Y.sub.b], b [member of] B such that [y.sub.1] [not equal to] [y.sub.2]. As X is FFSPS, there exists a proximal neighbourhood W>> b and a p-continuous map [alpha]: [X.sub.w] [right arrow] I such that [alpha]([y.sub.1]) = 0 and [alpha]([y.sub.2]) = 1, where [X.sub.w] [subset or equal to] X.

Obviously, [beta]: [X.sub.w] [intersection] Y [right arrow] 1 is a map in Y such that [beta]([y.sub.1]) = 0 and [beta]([y.sub.2]) = 1.

Hence subspace of fibrewise functionaly separated proximity space are fibrewise functionally separated.

4.11 Theorem: Let [phi]: X [right arrow] Y be p-continuous, fibrewise injective map, where W and Y are fibrewise proximity spaces over B. If Y is fibrewise functionally separated. Then so is X.

Proof: Let [x.sub.1], [x.sub.2] [member of] [X.sub.b], b [member of] B such that [x.sub.1] [not equal to] [x.sub.2]. Then [phi]([x.sub.1]), [phi]([x.sub.2]) [member of] [Y.sub.b] and [phi]([x.sub.1]) [not equal to] [phi]([x.sub.2]) i.e. [phi]([x.sub.1])[delta][phi]([x.sub.2]). Since Y is fibrewise functionally separated. Therefore, there exists a proximal neighbourhood W >> b and a p-continuous map [alpha]: [Y.sub.w] [right arrow] I such that

[alpha]([phi]([x.sub.1])) = 0 and [alpha]([phi]([x.sub.2])) = 1

[??]([alpha]0[phi])([x.sub.1]) = 0 and ([alpha]0[phi])([x.sub.2]) = 1 [??][beta]([x.sub.1]) = 0 and [beta]([x.sub.2]) = 1,

where [beta]: [X.sub.w] [right arrow] I is p-continuous.

Thus X is fibrewise functionally separated.

4.12 Definition: The fibrewise proximity space X over b is fibrewise regular if for each point x [member of] [X.sub.b], b [member of] B and for each proximal neighbourhood V >> x in X, there exists a proximal neighbourhood W >> b in B and a proximal neighbourhood U >> x in [X.sub.w] such that [X.sub.w] [intersection] [bar.U] << V.

4.13 Theorem: Let [phi]: X [right arrow] X' be a fibrewise embedding, where X and X' are fibrewise proximity space over B. If X' is fibrewise regular then so is X.

Proof: Let x [member of] [X.sub.b], b [member of] B and let V >> xin X.

Then V = [[phi].sup.-1]V', where V' >> x' = [phi](x) in X'.

Since X' is fibrewise regular there exists a proximal neighbourhood W >> b and a proximal neighbourhood U' >> x' in X 'w such that [X'.sub.w] [intersection] [bar.U'] << V'.

Then U = [[phi].sup.-1]U' >> x in [X.sub.w] such that [X'.sub.w] [intersection] [bar.U] << V. Thus X is fibrewise regular.

4.14 Theorem: Let [phi]: X [right arrow] Y be an p-open, p-closed and p-continuous fibrewise onto map, where X and Y are fibrewise proximity space over B. If X is fibrewise regular then so in Y.

Proof: Let y [intersection] [Y.sub.b], b [intersection] B and let V >> y.

Since [phi] is onto. Choose x [intersection] [[phi].sup.-1](y).

Then U = [[phi].sup.-1](V) >> x

By hypothesis, X is fibrewise regular there exists a proximal neighbourhood W >> b and a proximal neighbourhood U' >> x such that [X.sub.w] [intersection] U' << U.

Since [phi] is closed.

Therefore, [phi]([X.sub.w] [intersection] [bar.U']) << [phi](U)

[??]([X.sub.w]([intersection][phi]([bar.U']) << V

[??][Y.sub.w] [intersection][phi]([bar.U']) << V

[??][Y.sub.w] [intersection][MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] << V (since [phi][bar.U'] = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

Thus Y is fibrewise regular.

4.15 Theorem: Let X be fibrewise regular and fibrewise T over B. Then X is fibrewise Hausdorff.

Proof: Let [x.sub.1], [x.sub.2] [member of] [X.sub.b], b [member of] B and [x.sub.1] [not equal to] [x.sub.2]. Since X is fibrewise To, there exists a proximal neighbourhood say G of [x.sub.1], which does not contain [x.sub.2] i.e. [x.sub.1] << G, [x.sub.2] [not member of] G or [x.sub.2][delta]X -- G.

Also, since X is fibrewise regular, there exists a proximal neighbourhood W>> b in B and a proximal neighbourhood U >> [x.sub.1] in [X.sub.w] such that [X.sub.w] [intersection] [bar.U] << G.

Then U and [X.sub.w] [intersection] [X.sub.w] [??] are disjoint proximal neighbourhood of [x.sub.1] and [x.sub.2].

Hence X is fibrewise Hausdroff.

4.16 Definition: Let X be a fibrewise proximity space. Then x is said to be a fibrewise completely regular if for each x [member of] [X.sub.b], b [member of] B and a proximal neighbourhood V of x, there exists a proximal neighbourhood W >> b and a p-continuous map [p.sub.o]: [X.sub.w] [right arrow] I such that [p.sub.o](x) = 1 and [p.sub.o](X-V) = 0.

4.17 Theorem: Let [phi]: X [right arrow] X' be a fibrewise embedding, where X and X' are fibrewise proximity spaces over B. If X' is fibrewise completely regular, then so is X.

Proof: Let x [member of] [X.sub.b], b [member of] B and a proximal neighbourhood V >> x in X. Then V = [[phi].sup.-1]V', where V' >> x' = [phi](x) in X'.

Since X' is fibrewise completely regular, there exists a proximal neighbourhood W >> b in B and a p-continuous map [p.sub.o]: [X.sub.w] [right arrow] I such that [p.sub.o](x') = 1 and [p.sub.o](X-V') = 0.

Then U = [[phi].sup.-1]U' >> x in [X.sub.w] such that

[p.sub.o](x) = 1 and [p.sub.o](X-U) = 0.

Hence X is fibrewise completely regular.

4.18 Theorem: Let [phi]: X [right arrow] Y be a p-open, p-closed and p-continuous fibrewise onto map, where X and Y are fibrewise proximity space over B. If X is fibrewise completely regular, then so is Y.

Proof: Let y [member of] [Y.sub.b], b [member of] B and let V >> y in Y. As [phi] is onto. Choose x [member of] [[phi].sup.-1](y).

Since [phi] is p-continuous map. Therefore, x << [[phi].sup.-1](V) = U in X.

By hypothesis, X is fibrewise completely regular proximity space. Therefore, there exists a proximal neighbourhood W >> b in B and continuous function [p.sub.o]: [X.sub.w] [right arrow] I such that [p.sub.o](x) = I and [p.sub.o](X-U) = 0.

Using 3.4, we obtain a p-map [beta]: [X.sub.w] [right arrow] I such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

= sup{[p.sub.o](x): x [member of] [[phi].sup.-1](y)} = 1

[beta](Y-V) = sup{[p.sub.o](x): x [member of] [[phi].sup.-1](Y-V)}

= sup{[p.sub.o](x): x [member of] X-U} = 0.

Hence Y is fibrewise completely regular.

4.19 Theorem: Let X be a fibrewise completely regular and fibrewise To over B. Then X is fibrewise functionally Hausdorff.

Proof: Let [x.sub.1], [x.sub.2] [member of] [X.sub.b], b [member of] B and [x.sub.1] [not equal to] [x.sub.2].

Since X is fibrewise To, there exists a proximal neighbourhood say G of [x.sub.1] which does not contain [x.sub.2] i.e. << G, [x.sub.2] [not member of] G or [x.sub.2][delta]X-G.

Also, since X is fibrewise completely regular proximity space, there exists a proximal neighbourhood W of b in B (W >> b) and a continuous map [p.sub.o]: [X.sub.w] [right arrow] I such that [p.sub.o]([x.sub.1]) = 1 and [p.sub.o](X-G) = 0. Since [x.sub.2] [member of] X-G, it follows that [p.sub.o]([x.sub.2]) = 0.

Thus, X is fibrewise functionally Hausdroff.

4.20 Theorem: Let [phi]: X [right arrow] Y be a p-open, p-closed and p-continuous fibrewise onto map, where X and Y are fibrewise proximity space over B. If X is fibrewise completely regular and fibrewise. To, then Y is fibrewise functionally Hausdroff.

Proof: Let [y.sub.1], [y.sub.2] [member of] [Y.sub.b], b [member of] B such that [y.sub.1] [not equal to] [y.sub.2].

Since [phi] is onto map.

Choose [[phi].sup.-1]([y.sub.1]), [[phi].sup.-1]([y.sub.2]) [member of] [X.sub.b] and [[phi].sup.-1]([y.sub.1]) [not equal to] [[phi].sup.-1]([y.sub.2]).

By hypothesis, X is fibrewise functionally Hausdroff. Therefore, there exists a proximal neighbourhood W >> b in B and continuous function [alpha]: [X.sub.w] [right arrow] I such that

[alpha]([[phi].sup.-1])([y.sub.1])) = 0 and [alpha]([[phi].sup.-1])([y.sub.2])) = 1

[??]([alpha]o[[phi].sup.-1])([y.sub.1]) = 0 and ([alpha]o[[phi].sup.-1])([y.sub.2]) = 1

[??][beta]([y.sub.1]) = 0 and [beta]([y.sub.2]) = 1 {since [alpha] and [phi] both are p-continuous map}.

Where [beta]: [X.sub.w] [right arrow] I is p-continuous map.

Hence Y is fibrewise functionally Hausdroff.

4.21 Definition: Let X be a FPS. Then X is said to be a fibrewise normal proximity space (FNPS) if for each b e B and each pair of sets A and B of X such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], there exists a proximal neighbourhood W >> b and proximal neighbourhoods C and D of A and B respectively, such that [X.sub.w] [intersection] A << C and [X.sub.w] [intersection] B << D in [X.sub.w].

4.22 Theorem: Let [phi]: X [right arrow] Y be a, p-closed and p-continuous fibrewise onto map, where X and Y are fibrewise proximity spaces over B. If X is fibrewise normal, then so in Y.

Proof: Let b e B and let U and V be disjoint closed sets of Y. Then [[phi].sup.-1]U and [[phi].sup.-1]V are disjoint closed sets of X. Since X is fibrewise normal, there exists a proximal neighbourhood W of b and proximal neighbourhoods G and H of [[phi].sup.-1]U and [[phi].sup.-1]V respectively, such that [X.sub.w] [intersection] [[phi].sup.-1]U << G,

[X.sub.w] [intersection] [[phi].sup.-1]v << h.

[??] [X.sub.w] [intersection] [[phi].sup.-1]U[delta]([X.sub.w]-G), [X.sub.w] [intersection] [[phi].sup.- 1]V[delta]([X.sub.w]-H).

Since [phi] is closed.

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Thus Y is fibrewise normal.

4.23 Theorem: [phi]: X [right arrow] X' is a p-lcosed fibrewise embedding, where X and X' are fibrewise proximity spaces over B. If X' is fibrewise normal then so is X.

Proof: Let b [member of] B and let U, V be disjoint closed sets of X.

Then [phi]U, [phi]V are disjoint closed sets of X'.

Since X' is fibrewise normal, therefore, there exists a proximal neighbourhood W of b and proximal neighbourhoods G and H of [phi]U and [phi]V respectively, such that [X'.sub.w] [intersection] [phi]U << g, [X'.sub.w] [intersection] [phi]V << H.

Since [phi] is closed. Therefore,

[[phi].sup.-1]([X'.sub.w] [intersection] [phi]U) << [[phi].sup.-1]([X'.sub.w] [intersection] [phi]v) << [[phi].sup.-1]H [??] U [intersection] [X.sub.w] << [[phi].sup.-1]G, V [intersection] [X.sub.w] << [[phi].sup.-1]H.

Thus X is fibrewise normal.

References

[1] James I.M, Fibrewise Topology, Cambridge University Press (1990).

[2] James I.M., Uniform spaces over a base, J.London Math.Soc. (2), 32 (1985), 328-36.

[3] Naimpally, S.A. and Warrack, B.D., Proximity Spaces, Cambridge University Press (1970).

Sweety Agrawal and Sangita Srivastava

Department of Mathematics, H.C.P.G. College, Varanasi, Uttar Pradesh, India

Corresponding Author E-mail: sweety_agrawal30@yahoo.com
COPYRIGHT 2010 Research India Publications
No portion of this article can be reproduced without the express written permission from the copyright holder.