Arenas F - Alexandrov Spaces.pdf - Topologia i Geometria - renvox

Acta Math. Univ. Comenianae

Vol. LXVIII, 1(1999), pp. 17–25

ALEXANDROFF SPACES

F. G. ARENAS

Abstract. In this paper we mean by an Alexandro space a topological space such

that every point has a minimal neighborhood. We do not assume that the space

is T 0 . There spaces were rst introduced by P. Alexandro in 1937 in [1] and have

become relevant for the study of digital topology. We make a systematic study of

them from several points of view, including quasi-uniform spaces.

1. Introduction

In this paper we mean by an Alexandro space (or an space with the property

of Alexandro) a topological space such that every point has a minimal neighbor-

hood, or equivalently, has unique minimal base. This is also equivalent to the fact

that the intersection of every family of open sets is open. The minimal neighbor-

hood is denoted by V (x) and is the intersection of all open sets containing x. We

do not assume that the space is T 0 .

Although they were rst introduced by P. Alexandro in 1937 in [1] with the

name of Diskrete Raume, these spaces have not between systematically stud-

ied, perhaps because there were no reasons to do it. In fact the only references

that the author was able to nd before the eighties were two ([8] and [7]) that

are mainly concerned with nite spaces (the most important particular case, un-

doubtely) and make claims (easily proved) of the type this property also hold

for Alexandro spaces.

In the eighties, the interest in Alexandro spaces was a consequence of the

very important role of nite spaces in digital topology and the fact that these

spaces have all the properties of nite spaces relevant for such theory (see [6], [5]).

However nobody has intended (as far as I know) a systematic study of all topo-

logical properties of these spaces, independently the property has direct digital

applications or not. That is the main purpose of this paper.

Received July 18, 1996.

1980 Mathematics Subject Classication (1991 Revision). Primary 54F05; Secondary 54D10,

54E15, 54E05.

Key words and phrases. Alexandro spaces, quasi-uniform spaces, homotopy, space of

functions.

The author is partially supported by DGES grant PB95-0737.

F. G. ARENAS

2. Topological Properties of Alexandroff Spaces

We begin with the following characterization of a minimal base.

Theorem 2.1. Let X be an Alexandro space and U a family of open sets.

Then U is the minimal base for the topology of X if and only if:

1. U covers X.

2. If A,B 2U there exists a subfamily {U i : i 2 I} of U such that A\B =

S i2I U i .

3. If a subfamily {U i : i 2 I} of U veries S i2I U i 2 U, then there exists

i 0 2 I such that S i2I U i = U i 0 .

We show now in the next result how minimal bases can be induced in a subspace

or in a product, so the Alexandro property is hereditary and nitely productive.

Theorem 2.2. Let X and Y be Alexandro spaces with minimal bases U and V.

Then

1. If X is a subspace of Y , then U = {V \X : V 2V}.

2. X ×Y is an Alexandro space with minimal base U×V = {U ×V : U 2

U, V 2V}.

These rst two results are quoted without proof from [8]. Note that the property

of Alexandro is not countably productive, since discrete spaces are Alexandro

and any compact metric totally disconnected space is a subspace of a countable

product of nite discrete spaces (see [9, 29.15]) and the Cantor set is not Alexan-

dro, for example.

The following result comes from [1].

Theorem 2.3. Let X be an Alexandro space. X is T 0 if and only if V (x) =

V (y) implies x = y.

Note that if X is Alexandro, X is T 1 if and only if V (x) = {x}, and in

that case the space is discrete. The interest of this result (and the reason to

impose the hypothesis of being T 0 in the rest of the paper) is that allows to

consider a functional equivalence between the categories of T 0 Alexandro spaces

and partially ordered sets (posets in the following):

Given a poset P we construct the T 0 -Alexandro space X(P) as the set P

with the topology generated by {] ,x] : x 2X}, which is a T 0 -Alexandro space

with V (x) = ] ,x]. Conversely, given a T 0 -Alexandro space X, we construct

a poset P(X) as the set P with the order x y if and only if x 2 V (y). It

is straightforward that X(P(X)) = X and P(X(P)) = P and that under the

functors, continuous mappings become order preserving mappings and conversely

(see Section 4 of [7]). Note that the orders can be dened in the reversed way.

There is another way to assign a topological space to a poset, via the geometric

realization of a simplicial complex, as can be found in Section 9 of [2]. That is,

ALEXANDROFF SPACES

given a poset P, the order complex (P) of P is the simplicial complex whose

k-faces are k-chains in P. Conversely, given a simplicial complex K the face poset

P(K) is K ordered by inclusion. There is also a standard way to topologize a

simplicial complex called the geometric realization. We denote it by |K|.

In Section 2 of [7], the simplicial complex (P(X)) is called barycentric sub-

division, so we are going to denote it by sd X; on the other hand it is customary

(see again Section 9 of [2]) to call sd K = (P(K)) the barycentric subdivision of

the simplicial complex K. There is no confusion between the two notations.

Now, to relate topological properties of the space |sd (X)| (what is called the

geometric realization of X) with those of X, we quote with our notation Theo-

rems 2 and 3 of [7].

Theorem 2.4. There exists a weak homotopy equivalence f X : |sd (X)| ! X

dened as f X (u) = x 0 where u is in the unique open simplex of |sd (X)| with

vertices {x 0 ,...,x n }2 sd (X) and x 0 < ···< x n in P(X).

Each mapping : X ! Y between T 0 -Alexandro spaces induces a simplicial

mapping | |: |sd (X)|!|sd (Y )| such that f X = f Y | |.

Theorem 2.5. There exists a weak homotopy equivalence g K :|K|!X(P(K)),

dened as g K = f X(P(K)) , since (P(X(P(K)))) = sd K and |sd K| = |K|.

Each simplicial mapping : K ! L between simplicial spaces induces a mapping

: X(P(K)) !X(P(L)) such that f K is homotopic to f L | |.

The following result shows the behaviours of these functors.

Theorem 2.6. Let X and Y be T 0 -Alexandro spaces and let P and Q be

posets.

1. (P(sd X)) = sd (P(X)) and (P(sd n X)) = sd n (P(X)).

2. P( · i2I X i ) = · i2I P(X i ).

3. P(X × Y ) = P(X) × P(Y ) (× is the direct product between the posets

P(X) and P(Y )).

Proof. Straightforward.

€

Note that X(P ×Q) is P ×Q topologized by V (x,y) = V (x)×Q[{x}×V (y),

so is not X(P) ×X(Q).

Now we shall study the connectivity properties of these spaces.

Theorem 2.7. Let X be a T 0 -Alexandro space. The following statements are

equivalent.

1. X is path-connected.

2. X is connected.

3. X is chain-connected.

4. For every a,b 2X, there exist a 0 ,...,a n+1 2X such that a 0 = a, a n+1 =

b and V (a i )\V (a j ) 6= ; if |i−j| 1.

F. G. ARENAS

5. For every a,b2 X, there exist a 0 ,...,a m+1 2 X such that a 0 = a, a m+1 =

b and V (a i )\V (a j ) 6= ; if |i−j| 1.

6. For every a,b 2X, there exist a 0 ,...,a k+1 2X such that a 0 = a, a k+1 =

b and {a i }\{a j }6= ; if |i−j| 1.

Proof. (1) ) (2) ) (3) are valid in any topological space, (4) is the form that

(3) has in these spaces and (4) ) (5) ) (6) ) (1) is straightforward. Note that

if we rewrite (4) by means of the poset P(X) we obtain [8, 3.5]. €

The following results are an account of point-set topological properties satised

by T 0 (and non-T 0 ) Alexandro spaces.

Theorem 2.8. Let X be a T 0 -Alexandro space.

1. X is locally path-connected.

2. X is rst countable.

3. X is orthocompact.

4. X is paracompact if and only if every V (x) meets only a nite number of

V (y), so if X is paracompact, them X is locally nite (but not conversely).

5. X is second countable if and only if it is countable.

6. X is separable if and only if X = S n=1 {x n }.

7. X is Lindelof if and only if X = S n=1 V (x n ).

8. There exists Lindelof T 0 -Alexandro spaces that are not separable and

separable T 0 -Alexandro spaces that are not Lindelof.

9. If X is nite, then X is compact.

10. If X is locally nite, then it is locally compact.

11. X is countable if and only if X is locally countable and Lindelof.

12. If X is locally nite, X is compact if and only if X is nite.

Proof. 1. Apply the preceding theorem to the minimal neighborhood.

2. Obvious,

3. The minimal base is an open interior/preserving renement of every open

covering.

4. The minimal base is a renement of every open cover, so the rst assertion

is clear from the denition of paracompactness.

To prove assertion, suppose not, that is, there exists x 2 X such that V (x) is

innite. Then V (y) V (x) for every y 2 V (x), a contradiction. X = N with

V (n) = {1,...,n} is locally nite (and countable) but not paracompact, since

V (1) meets every V (n).

5. Countable and rst countable, in any topological space, implies second count-

able. Since second countable mean that the minimal base is countable, so is the

space.

6. If D is a countable dense subset, D = {x n : n 2 N} and for every x 2 X,

V (x) \D is nonempty, hence x n 2 V (x) for some n 2 N, what means x 2{x n },

so X = S n=1 {x n }. The converse is in the same way.

ALEXANDROFF SPACES

7. Apply that X is Lindelof to the covering {V (x) : x 2 X}. For the converse,

if U is an open covering, for each U 2 U there exists n 2 N such that x n 2 U.

Rename that U as U n and note that V (x n ) U n .

8. If X = R

X : y x}, X = {0}, so X is separable, but S n=1 V (x n ) = [0, Sup{x n : n 2 N}]

and Sup{x n : n 2N} = < ! 1 (the countable supremum of countable ordinals is

countable).

9. Obvious.

10. Obvious.

11. From (7) X is a countable union of countable subsets.

12. Apply compactness to the covering {V (x) : x 2X}.

€

Theorem 2.9. Let X be an Alexandro space.

1. X is regular if and only if V (x) is closed for every x 2 X (hence X is

0-dimensional).

2. If X is regular and compact, then X is locally compact.

3. If X is regular and separable, then X is perfectly normal.

4. X is pseudo-metrizable if and only if V (x) is closed and nite for every

x 2X.

Proof. 1. X is regular means X locally closed.

2. Obvious.

3. Note rst that if X is regular, from (1) we have that {x n } V (x n ) and (6)

and (7) of 2.8 and the separability of X give X is Lindelof. Recall also that if X

is regular and Lindelof, it is also normal. As in (6) of 2.8, we can write any open

set A as A = S n=1 {x n } (the fact that {x n } A comes from regularity), so every

open set is F ; this together with the normality gives perfect normality.

4. From the Nagata-Smirnov’s pseudo-metrization theorem, X is semimetriz-

able if and only if X is regular and has a -locally nite base. Since a theorem of

Michael says that a space is paracompact if and only if every open covering has a

-locally nite renement, with a reasoning similar to that of (3) of 2.8 we have

that X has a -locally nite base if and only if X is paracompact. But under the

additional condition of being V (x) closed for every x2X we have that paracom-

pact is equivalent to locally nite (V (y) \V (X) 6= ; means y 2 V (x) = V (x), so

the set of y 0 s such that V (y) meets V (x) is the set of elements of V (x)). €

Note that the only T 0 -Alexandro pseudo-metrizable spaces are the discrete

ones (which are metrizable in fact).

0 and V (0) = X, V (x) = {x} for every x > 0, we have that

{0} = {0} and {x} = {0,x} for every x > 0, so X is Lindelof but not separable.

On the other hand, if X = [0,! 1 [, the rst uncountable ordinal and V (x) = {y 2

Arenas F - Alexandrov Spaces.pdf

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: