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Unit 24: Local Finiteness and Paracompactness

 A collection  of subsets of X is said to be countably locally finite is  can be written as the Notes
countable union of collections  , each of which is locally finite.
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 Let  be a collection of subsets of space X. A collection  of subsets of X is said to be a
refinement of  if for each element B of , there is an element A of  containing B. If the
elements of  are open sets, we call  an open refinement of ; if they are closed sets, we
call  a closed refinement.
 A space X is paracompact if every open covering  of X has a locally finite open refinement
 that covers X.

24.4 Keywords

Metrizable: Any topological space (X, T) if it a possible to find a metric  on X which induces the
topology T i.e. the open sets determined by the metric are precisely the members of T, then X
is said to the metrizable.
Open Cover: Let (X, T) be a topological space and A X let G denote a family of subsets of X. G
is called a cover of A if A  {G : G G}.

24.5 Review Questions

1. Give an example to show that if X is paracompact, it does not follow that for every open
covering  of X, there is a locally finite subcollection of  that covers X.
2. (a) Show that the product of a paracompact space and a compact space is paracompact.
[Hint: Use the tube lemma.]

(b) Conclude that S is not paracompact.

3. Is every locally compact Hausdorff space paracompact?
4. (a) Show that if X has the discrete topology, then X is paracompact.
(b) Show that if f : X Y is continuous and X is paracompact, the subspace f(X) of Y need
not be paracompact.

5. (a) Let X be a regular space. If X is a countable union of compact subspace of X, then X is
paracompact.
(b) Show  is paracompact as a subspace of w in the box topology.
6. Let X be a regular space.
(a) Show that if X is a finite union of closed paracompact subspaces of X, then X is
paracompact.
(b) If X is a countable union of closed paracompact subspaces of X whose interiors cover
X, show X is paracompact.
7. Find a point-finite open covering  of that is not locally finite (The collection  is point
finite if each point of lies in only finitely many elements of ).
8. Give an example of a collection of sets  that is not locally finite, such that the collection

 = {A/A  } is locally finite.
9. Show that if X has a countable basis, a collection  of subsets of X is countably locally
finite if and only if it is countable.
10. Consider w in the uniform topology. Given n, let  be the collection of all subsets of w
n
of the form A ; where A = for i n and A equals either {0} or {1} otherwise. Show that
i i i
collection  =  is countably locally finite, but neither countable nor locally finite.
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