Download Problem Sheet 3

Problem Sheet 3
November 2, 2016
1. Paracompactness
(a) Suppose A is alocally finite
family of subsets of a topological space
X. Show that A : A ∈ A is a locally finite family as well.
(b) Suppose X is regular. Show that if every open cover U of X has
a locally finite refinement A covering X, then for every open cover
U there is a closed locally finite cover {FU : U ∈ U } of X such that
∀U ∈ U FU ⊆ U .
(c) Show that if X is a topological space then every open σ-locally finite
cover of X has a locally finite refinement covering X.
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Hint: If Un is an open cover with each Un being locally finite, try
to build the locally finite refinement in stages along n, making sure
that each point is covered ‘as little as possible’.
(d) Complete the proof of Lemma 7.9: Suppose X is a regular space.
TFAE:
i. X is paracompact
ii. Every open cover of X has a σ-locally finite open refinement
covering X.
iii. Every open cover of X has a locally finite refinement covering
X.
iv. Every open cover of X has a locally finite closed refinement covering X.
Hint: For (iv) implies (i), take an open cover U , apply (iv) to obtain a locally finite closed refinement C covering X, take a cover W
witnessing local finiteness of C and a locallySfinite closed refinement
D of W covering X. For C ∈ C let VC = X \ {D ∈ D : D ∩ C = ∅},
show that the VC form a locally finite open cover of X and use it to
obtain a locally finite open refinement of U .
(e) Suppose X is regular. Show (by modifying the proof from (b) or
otherwise) that if X is paracompact then for every open cover U there
is a locally finite open cover {VU : U ∈ U } such that ∀U ∈ U VU ⊆ U .
(f) This generalizes that separable metric spaces are Lindeloef: Show
that every separable paracompact regular space is Lindeloef.
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2. Convergence and Filters
(a) Check all the unproven statements about filters and convergence from
the lecture notes. In particular:
i. x ∈ A iff there is a filter F ∋ A converging to x iff there is an
ultrafilter F ∋ A converging to x.
ii. If f : X → Y , F is a filter on X and G is the filter generated by
f (F) then A ∈ G ⇐⇒ f −1 (A) ∈ F and if f is surjective then
f (F) is a filter.
iii. If f : X → Y is a continuous function between topological spaces
then f is continuous iff f preserves limits of filters iff f preserves
limits of ultrafilters. [Hint: Use f is continuous iff for each
A ⊆ X, f (A) ⊆ f (A).]
iv. X is Hausdorff iff every filter has at most one limit point.
(b) Show that if X is a topological space then for all filters F, G and
points x ∈ X
i.
ii.
iii.
iv.
v.
F ⊆ G =⇒ lim F ⊆ lim G;
F → x if and only if for every ultrafilter U ⊇ F we have U → x;
Px = {A ⊆ X : x ∈ A} → x;
if F, G → x then F ∩ G → x (and F ∩ G is a filter);
the previous statement remains true for infinitely many filters
converging to x;
(c) Optional/Revision: Give an example, showing that if X is a set,
→ is a relation between filters on X and points of X satisfying all the
conditions of the previous part then c(A) = {x ∈ X : ∃ filter F ∋ A : F → x}
might not a closure operator (in particular c(c(A)) 6= c(A) in general).
(d) Suppose now that → is a relation between filters on X and points
of X satisfying all the conditions of the part above. Show that τ =
{U ⊆ X : ∀x ∈ U ∀ filter F → x U ∈ F} is a topology on X. Show
that if → is the convergenc relation of some topology σ on X then
τ = σ. Optional: Give an example showing that the convergence
relation of τ might not coincide with →.
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(e) Show that if U is a non-principal ultrafilter if and only if U = ∅.
These ultrafilters are called free ultrafilters. Show that an ultrafilter
U is free if and only if it contains the cofinite filter.
(f) Show that there no free ultrafilter with a countable filter basis.
(g) Show that there is no countably complete (i.e. closed under countable
intersections) free ultrafilter on [0, 1].
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