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 is a second countable space. If a non-empty
subset
is covered by a class of open
sets then can be covered by a countable subcover.
Problem 2: is second countable.
Prove any open base for 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, ,
with
not Lindelöf).
(*) Problem 7: Is the product of second countable spaces
second countable (product means Tychonoff product)?
Problem 8: What is the weakest topology?
Problem 9: Is
a subspace of
?