Revista Matematica

Appendix
A Primer on Topological Spaces

This appendix aims to provide a succinct, and yet self-contained, discussion of some basic elements of general topology and metric analysis. We concentrate here only on those aspects of topological notions and metric space concepts that are relevant for the present coverage of order theory. Furthermore, for brevity, we leave the proofs of most of the(easy) propositions as exercises. All major theorems that are essential to the present discussion are proved, however.

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1.1

Topological Spaces
Basic De…nitions

We begin with the axiomatic de…nition of the notion of “topological space.” De…nition. Let X be a nonempty set. A topology on X is a collection OX of subsets of X such that ; 2 OX and X 2 OX ; U 2 OX for any nonempty subset U ofOX ; and U 2 OX for any nonempty …nite subset U of OX :

T

S

Formally speaking, a topological space is a list (X; OX ) where X is a nonempty set and OX is a topology on X: However, in practice, we omit giving speci…c reference to the involved topology, and refer to X itself as a topological space. We shall denote the topology of an arbitrarily given topological space X as OX in whatfollows. In addition, we refer to every member of OX as open (in X). An open 1

subset of X that contains an element x of X is, in turn, called an open neighborhood of x: More generally, a neighborhood of x is a subset of X that contains an open neighborhood of x: Finally, we refer to any subset S of X such that XnS is open as closed. The collection of all closed subsets of X is denoted as CX :Finally, any subset of X that is both open and closed, such as ; and X; is called clopen.
Note. CX is nonempty for any topological space X, for it surely contains ; and X: This collection is closed under arbitrary intersections (that is, the intersection of any collection of closed sets is closed), and it is closed under …nite unions (that is, the union of …nitely many closed sets is closed).Example 1.1.1. Let X be a nonempty set. Then, 2X and f;; Xg are topologies on X: We say that X is a discrete space if it is endowed with the topology 2X ; and that it is an indiscrete space if it is endowed with the topology f;; Xg: In a discrete space, every set is open. (In particular, in this case, fxg is an open neighborhood of x for any x 2 X.) This, in turn, implies that every subset of a discretespace is closed as well. By contrast, the only open (and hence closed) subsets of an indiscrete space X is ; and X: Let X be a topological space, and S a subset of X: The interior of S (in X) – denoted as int(S) –is the -maximum open set in X that is contained in S: It is easy to see that this means [ int(S) = fO 2 OX : O Sg; which, in particular, shows that S is open i¤ int(S) = S: Similarly,the closure of S (in X) –denoted as cl(S) –is the -minimum closed set in X that contains S: It is easy to see that this means \ cl(S) = fC 2 CX : S Cg; which, in particular, shows that S is closed i¤ cl(S) = S: (The set cl(S)nint(S) is called the boundary of S:) Finally, we say that S is dense in X; if cl(S) = X: (Dense sets are thought of as “large”from the topological point of view.) The followingproposition, whose proof is easy, gives an alternative characterization of the closure of a set.

Proposition 1.1.1. Let X be a topological space, and S a subset of X: Then, an element x of X belongs to cl(S) if, and only if, every open neighborhood of x intersects S: As an immediate corollary we get: 2

Corollary 1.1.2. A subset S of a topological space X is dense if, and only if, everyopen set in X intersects S:

We introduce next a very important class of topological spaces.

De…nition. We say that a topological space X (or the topology of that space) is Hausdor¤, if for any distinct elements x and y of X; there exist disjoint open subsets O and U of X such that x 2 O and y 2 U: It is easily checked that every singleton, and hence every …nite set, in a Hausdor¤ topological...