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hw4
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One-page homework assignment from MATH 4181 section 001, Fall 1999, in a folder of Royster topology course notes. Problems cover closed sets in a topological space, discrete spaces, the finite complement and countable complement topologies, limit points and derived sets, the Sierpinski space, and counting topologies on a three-element set. Two exercises are cited from pages 74-75 of the course textbook.
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MATH 4181 001 Fall 1999
Problem Set 4
1. (Exercise 4, page 74) Let ( X;T) be a topological space. Prove that ;andXare closed
sets, that a nite union of closed sets is a closed set, and that an arbitrary intersection
of closed sets is a closed set.
2. (Exercise 6, page 75) Prove that in a discrete topological space, each subset is simul-
taneously open and closed.
3. Show that a topological space ( X;T) is discrete if and only if each set consisting of
only one point is open.
4. LetXbe a set and let T0the nite complement topology for X.
(a) Show that ( X;T0) is discrete if and only if Xis a nite set.
(b) Show that if Ais an in nite subset of X, then every point of Xis a limit point
ofA.
5. LetXbe a set. The countable complement topology , orco-countable topology ,T00for
Xconsists ofX,;and all subsets OofXfor whichXnOis a countable set.
(a) Show that T00is a topology on X.
(b) For the space ( X;T00), show that a countable set AofXhas a derived set A0=;
and that an uncountable set BhasB0=X.
(c) Show that the intersection of any countable family of members of T00is a member
ofT00.
6. LetX=fa;bgbe a two-element set and let T=f;;fag;fa;bgg. Show that Tis a
topology on Xand identify the limit points of each subset of X. (This space is called
Sierpenski space .)
7. How many di erent topologies are there for a set with three members?
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