109

Thus

By (b), there is a member B ofx with

Then O1 . O is the union of such members B of2 x and is therefore a member of Thus is a

.topology for X

Example 4.3.4

Let be the family of all intervals in of the form [ , ), . It is easily observed that a b a < b satisfies the conditions of and is a basis for a topology, called the Theorem 4.8 half-open interval for . It is left as an exercise for the reader to show that is firsttopology countable and separable but not second countable. (The real line with the half-open interval topology is sometimes called the .)Sorgenfrey line Definition: . Let and be bases for topologies and for a set X Then S and are .equivalent bases provided that the topologies and are identical The proof of the following theorem is left as an exercise. Theorem 4.9: Bases and for topologies on a set X are equivalent if and only if both of the following conditions hold: (a) For each , .and x B there is a member such that x B B (b) For each , .and x B there is a member such that x B B In some instances it is advantageous to have a smaller collection of sets which generates a basis by the process of forming finite intersections. Such a family, called a , is defined as follows:subbasis Definition: . Let be a space A subcollection of is a subbasis or subbase for if the .family of all finite intersections of members of is a basis for Example 4.3.5 The collection of all open intervals of the form ( , ) and (–, ), , , is a subbasis for thea b a usual topology for . EXERCISE 4.3Co py ri gh t © 2 01 6. D ov er P ub li ca ti on s. A ll r ig ht s re se rv ed . Ma y no t be r ep ro du ce d in a ny f or m wi th ou t pe rm is si on f ro m th e pu bl is he r, e xc ep t fa ir u se s pe rm it te d un de r U. S. o r ap pl ic ab le co py ri gh t la w. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 9/26/2021 6:19 PM via AMERICAN PUBLIC UNIVERSITY SYSTEM AN: 1565225 ; Croom, Fred H..; Principles of Topology Account: s7348467 110 1. Let be a space and a subcollection of . Suppose that for each in , the set ofa X members of which contain is a local base at . Show that is a basis for .a a 2. Prove part (a) of .Theorem 4.7 3. Give an example different from of a space that is first countable but not second countable.Example 4.3.4 X 4. Let be a first countable space and a limit point of a subset of X Show that there is a sequence of points ofX x A { } which converges to .A x x 5. Describe the bases and for determined by the open balls of the taxicab metric and the maxd metric respectively. Show that and are both equivalent to the basis of open balls in thed usual metric .d 6. Let be a first countable space and a member of . Prove that there is a local nested basis at X x X a (i.e., a local basis such that for each positive integer ).Sn+1 Sn n 7. Let be the real line with the half-open interval topology of .Example 4.3.4 (a) Find the closure, interior, and boundary of the set = [0, 2] and the set = (0, 2).A B (b) Prove that is separable