TYPES OF TOPOLOGICAL VECTOR SPACES

A topological space is a set S in which a collection
r of subsets (called open sets) has been specified, with the following
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properties : S is open, 0 is open, the intersection of any two open sets is
open, and the union of every collection of open sets is open. Such a collection
r is called a topology on S. When clarity seems to demand it, the topological
space corresponding to the topology r will be written (S, r) rather
than S.
Here is some of the standard vocabulary that will be used, if S and r
are as above.
A set E c S is closed if and only if its complement is open. The closure E of E is the intersection of all closed sets that contain E. The interior Eo of
E is the union of all open sets that are subsets of E. A neighborhood of a
point p E S is any open set that contains p. (S, r) is a Hausdorff space, and r
is a Hausdorff topology, if distinct points of S have disjoint neighborhoods.
A set K c S is compact if every open cover of K has a finite subcover. A
collection r

c r is a base for r if every member of r (that is, every open set)
is a union of members of r

. A collection y of neighborhoods of a point
p E S is a local base at p if every neighborhood of p contains a member of y.
If E c S and if u is the collection of all intersections E n V, with
V E r, then u is a topology on E, as is easily verified ; we call this the topology
that E inherits from S.