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TOPOLOGY I, FALL 2003
HOMEWORK SET II


Problems labelled with (*) are to be handed in


Problems on 2nd countable spaces


Definition: A topological space, X. is second countable if it has a countable open base


Problem 1: Prove Lindelöf's theorem: Suppose $ X$ is a second countable space. If a non-empty subset $ G \subseteq X$ is covered by a class $ \{U_i\}$ of open sets then $ G$ can be covered by a countable subcover.


Problem 2: $ X$ is second countable. Prove any open base for $ X$ has a countable sub-base.


Define a space to be Lindelöf if every open cover has a countable subcover. Lindelöf's theorem says that a second countable space is Lindelöf.


(*) Problem 3: Is a Lindelöf space second countable?


The following problems are analogous to theorems on compact spaces.


Problem 4: Is the image of a Lindelöf space Lindelöf?


(*) Problem 5: Is any closed subspace of a Lindelöf space Lindelöf?


The main lemma that implies the Tychonoff theorem is the following: A topological space is compact if every subbasic open cover has a finite subcover.


(*) Problem 6 Is the above lemma true if compact is replaced by Lindelöf and finite is replaced by countable, i.e. ''A topological space is Lindelöf if every subbasic open cover has a countable subcover"? Prove it or provide a counter example. If you find a counter example, show how the proof fails. [HINT: It is a fact that the Tychonoff theorem does not generalize to Lindelöf spaces. I.E. the product of Lindelöf spaces (with the Tychonoff topology) may not be Lindelöf (see for example, Kelley, "General Topology" page 59, Problem L where there is an example of a Lindelöf space, $ X$, with $ X \times X$ not Lindelöf).


(*) Problem 7: Is the product of second countable spaces second countable (product means Tychonoff product)?


Problem 8: What is the weakest $ T_1$ topology?


Problem 9: Is $ {\Bbb R}^{\infty}$ a subspace of $ {\Bbb
R}^{ {\Bbb Z} }$?



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Martin Bendersky 2003-07-23