Metrization Theorems and Paracompactness

6 Metrization Theorems and Paracompactness

6.1 Local finiteness

6.1.1 Locally finite in X (a collection \mathcal{A} of subsets of a topological space X) : Every point of X has a neighborhood that intersects only finitely many elements of \mathcal{A}.

6.1.2 (Lem 39.1) Let \mathcal{A} be a locally finite collection of subsets of X. Then;

6.1.2.1 Any subcollection of \mathcal{A} is locally finite.

6.1.2.2 The collection \mathcal{B} = {\bar{A} }_{A \in \mathcal{A}} of the closures of the elements of \mathcal{A} is locally finite.

6.1.2.3 \bar{\bigcup_{A \in \mathcal{A}}} = \bigcup_{A \in \mathcal{A}} \bar{A}

6.1.3 Locally finite indexed family in X ( an indexed family {A_{\alpha}}{\alpha \in J} ) : every x \in X has a neighborhood that intersects A{\alpha} for only finitely many values of \alpha.

6.1.4 Countably locally finite ( a collection \mathcal{B} of subsets of X) : \mathcal{B} can be written as the countable union of collections \mathcal{B}_{n} , each of which is locally finite.

6.1.5 Refinement \mathcal{B} of \mathcal{A} ( a collection \mathcal{B} of subsets of X) : for each element B of \mathcal{B}, there is an element A of \mathcal{A} containing B.

6.1.5.1 Also called ‘ to refine \mathcal{A}’

6.1.5.2 Open refinement of \mathcal{A} ( a collection \mathcal{B} of subsets of X) : elements of \mathcal{B} are open sets

6.1.5.3 Closed refinement of \mathcal{A} ( a collection \mathcal{B} of subsets of X) : elements of \mathcal{B} are closed sets

6.1.6 (Lem 39.2) Let X be a metrizable space. If \mathcal{A} is an open covering of X, then there is an open covering \Epsilon of X that is countably locally finite.

6.1.6.1 Pf) (well-ordering theorem)

6.2 Nagata-Smirnov Metrization Theorem

6.2.1 G_{\delta} set in X ( a subset A of a space X) : it equals the intersection of a countable collection of open subsets of X.

6.2.2 (Lem 40.1) Let X be a regular space with a basis \mathcal{B} that is countably locally finite. Then X is normal, and every closed set in X is a G_{\delta} set in X.

6.2.2.1 Pf) (Lem 39.1) (Thm 32.1)

6.2.3 (Lem 40.2) Let X be normal; let A be a closed G_{\delta} set in X. Then there is a continuous function f : X-> [0,1] s.t. f(x) = 0 for x \in A and f(x) > 0 for x \notin A.\

6.2.4 (Nagata-Smirnov Metrization Theorem) (Thm 40.3) : A space X is metrizable iff X is regular and has a basis that is countably locally finite.

6.2.4.1 Pf) (Lem 39.2)

6.3 Paracompactness

6.3.1 Paracompact ( a space X) : every open covering \mathcal{A} of X has a locally finite open refinement \mathcal{B} that covers X.

6.3.1.1 \mathbb{R}^{n}

6.3.2 (Thm 41.1) Every paracompact Hausdorff space X is normal.

6.3.3 (Thm 41.2) Every closed subspace of a paracompact space is paracompact.

6.3.4 (Lem 41.3) Let X be regular. Then the following conditions on X are equivalent; Every open covering of X has a refinement that is;

6.3.4.1 An open covering of X and countably locally finite.

6.3.4.2 A covering of X and locally finite.

6.3.4.3 A closed covering of X and locally finite

6.3.4.4 An open covering of X and locally finite

6.3.4.5 Pf) (Lem 39.1)

6.3.5 (Thm 41.4) Every metrizable space is paracompact

6.3.5.1 Pf) (Lem 39.2)

6.3.6 (Thm 41.5) Every regular Lindelöf space is paracompact.

6.3.6.1 Pf) (Preceding lemma)

6.3.7 Partition of unity on X, dominated by {U_{\alpha}} ( a indexed open covering {U_{\alpha}}{\alpha \in J} of the space X) : Indexed family of continuous functions \phi{\alpha} : X -> [0,1] for i = 1, …, n if :

6.3.7.1 (support \phi_{\alpha}) \subset U_{\alpha} for each \alpha

6.3.7.2 The indexed family {Support \phi_{\alpha} } is locally finite

6.3.7.3 \sum \phi_{\alpha} (x) = 1 for each x.

6.3.8 (Shrinking lemma)(Lem 41.6) Let X be a paracompact Hausdorff space; Let {U_{\alpha}}{\alpha \in J} be an indexed family of open sets covering X. Then there exists a locally finite indexed family {V{\alpha}}{\alpha \in J} of open sets covering X s.t. \bar{V}{\alpha} \subset U_{\alpha} for each \alpha.

6.3.8.1 Family {\bar{V}{\alpha}} is a ‘precise refinement’ of the faminy {U{\alpha}} \bar{V}{\alpha} \subset U{\alpha} for each \alpha.

6.3.9 (Thm 41.7) Let X be a paracompact Hausdorff space; let {U_{\alpha}}{\alpha \in J} be an indexed open covering of X. Then there exists a partition of unity on X dominated by {U{\alpha}}.

6.3.9.1 Pf) (Shrinking lemma)

6.3.10 (Thm 41.8) Let X be a paracompact Hausdorff space; let \mathcal{C} be a collection of subsets of X; for each C \in \mathcal{C}, let \epsilon_{C} be a positive number. If \mathcal{C} is locally cinite, there is a continuous function f: X->\mathbb{R} s.t. f(x) >0 for all x, and f(x) \le \epsilon_{C} for x \in C.

6.4 Smirnov Metrization Theorem

6.4.1 Locally metrizable (a space X) : Every point x of X has a neighborhood U that is metrizable in the subspace topology.

6.4.2 (Smirnov metrization theorem) (Theorem 42.1) Space X is metizable iff it is a paracompact Hausdorff space that is locally metrizable.

6.4.2.1 Pf) similar to (Thm 40.3)