Characterization of Paracompactness in T3 Space

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Theorem

Let $T = \struct {X, \tau}$ be a $T_3$ space.


The following statements are equivalent:

$(1): \quad T$ is paracompact
$(2): \quad$ every open cover of $T$ has a locally finite refinement
$(3): \quad$ every open cover of $T$ has a closed locally finite refinement
$(4): \quad$ every open cover of $T$ is even
$(5): \quad$ every open cover of $T$ has an open $\sigma$-discrete refinement
$(6): \quad$ every open cover of $T$ has an open $\sigma$-locally finite refinement


Proof

Statement $(1)$ implies Statement $(2)$

Let $T$ be paracompact.


By definition of paracompact:

every open cover of $S$ has an open refinement which is locally finite.

By definition of open refinement:

every open refinement of a cover is a refinement of the cover.


It follows that:

every open cover of $T$ has a locally finite refinement.

$\Box$


Statement $(1)$ implies Statement $(6)$

Let $T$ be paracompact.


By definition of paracompact:

every open cover of $T$ has an open locally finite refinement


From Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets

every open cover of $T$ has an open $\sigma$-locally finite refinement

$\Box$


Statement $(2)$ implies Statement $(3)$

Let every open cover of $T$ have a locally finite refinement.


Let $\UU$ be an open cover of $T$.


Let $\VV = \set{V \in \tau : \exists U \in \UU : V^- \subseteq U}$ where $V^-$ denotes the closure of $V$ in $T$.

Lemma 1

$\VV$ is an open cover of $T$

$\Box$


By assumption:

there exists a locally finite refinement $\AA$ of $\VV$.

Let:

$\BB = \set{A^- : A \in \AA}$

From Closures of Elements of Locally Finite Set is Locally Finite:

$\BB$ is locally finite

Lemma 2

$\BB$ is a cover of $T$ consisting of closed sets

$\Box$

Lemma 3

$\BB$ is a refinement of $\UU$

$\Box$


Since $\UU$ was an arbitrary open cover of $T$ it follows that:

every open cover of $T$ has a closed locally finite refinement.

$\Box$


Statement $(3)$ implies Statement $(1)$

Let every open cover of $T$ have a closed locally finite refinement.


Let $\UU$ be an open cover of $T$.


Let $\VV$ be a closed locally finite refinement of $\UU$, which exists by assumption.


Let $\WW = \set{W \in \tau : \set{V \in \VV : V \cap W \ne \O} \text{ is finite}}$.


By definition of locally finite:

$\forall x \in X: \exists W \in \tau: x \in W$ and $\set{V \in \VV : V \cap W \ne \O}$ is finite.


Hence $\WW$ is an open cover of $T$, by definition.


Let $\AA$ be a closed locally finite refinement of $\WW$, which exists by assumption.

Lemma 4

$\forall A \in \AA : \set{V \in \VV : V \cap A \ne \O}$ is finite

$\Box$


For each $V \in \VV$, let:

$V^* = X \setminus \ds \bigcup \set{A \in \AA : A \cap V = \O}$

Lemma 5

$\forall V \in \VV: V \subseteq V^*$

$\Box$


Let $\VV^* = \set{V^* : V \in \VV}$.

Lemma 6

$\VV^*$ is an open locally finite cover of $T$

$\Box$


From Lemma 5 and Lemma 6 it follows that $\VV$ is a refinement of $\VV^*$ by definition.


By definition of refinement:

$\forall V \in \VV : \exists U \in \UU : V \subseteq U$


For each $V \in \VV$, let:

$U_V \in \UU : V \subseteq U_V$


Let:

$\UU^* = \set{V^* \cap U_V : V \in \VV}$

Lemma 7

$\UU^*$ is an open locally finite refinement of $\UU$

$\Box$


Since $\UU$ was arbitrary, it follows that $T$ is paracompact by definition.

$\Box$


Statement $(3)$ implies Statement $(4)$

This follows immediately from Open Cover with Closed Locally Finite Refinement is Even Cover.

$\Box$


Statement $(4)$ implies Statement $(5)$

Let every open cover of $T$ be even.


Let $\UU$ be an open cover of $T$.

Lemma 8

there exists a $\sigma$-discrete refinement $\AA$ of $\UU$

$\Box$


By definition of $\sigma$-discrete set of subsets:

$\AA = \ds \bigcup_{n \in \N} \AA_n$ where $\AA_n$ is a discrete set of subsets for each $n \in \N$.


Let $X \times X$ denote the cartesian product of $X$ with itself.


Let $\tau_{X \times X}$ denote the product topology on $X \times X$.


Let $T \times T$ denote the product space $\struct {X \times X, \tau_{X \times X} }$.

Lemma 9

Let $\BB$ be a discrete set of subsets of $X$.


Then there exists an open neighborhood $W$ of the diagonal $\Delta_X$ of $X \times X$ in $T \times T$:

$\forall x \in X : \card {\set{B \in \BB : \map W x \cap W \sqbrk B \ne \O}} \le 1$

$\Box$


From Lemma 9:

$\forall n \in \N$ there exists a neighborhood $V_n$ of the diagonal $\Delta_X$ of $X \times X$ in $T \times T$:
$\forall x \in X : \card {\set{A \in \AA_n : \map {V_n} x \cap V_n \sqbrk A \ne \O}} \le 1$


By definition of refinement:

$\forall A \in \AA : \exists U_A \in U : A \subseteq U_A$


For each $n \in \N$, let:

$\WW_n = \set{U_A \cap V_n \sqbrk A : A \in \AA_n}$


Let:

$\WW = \ds \bigcup_{n \in \N} \WW_n$

Lemma 10

$\WW$ is an open $\sigma$-discrete refinement of $\UU$

$\Box$


Since $\UU$ was arbitrary, then every open cover of $T$ has an open $\sigma$-discrete refinement

$\Box$


Statement $(5)$ implies Statement $(6)$

Follows immediately from Sigma-Discrete Set of Subsets is Sigma-Locally Finite.

$\Box$


Statement $(6)$ implies Statement $(2)$

Let every open cover of $T$ have an open $\sigma$-locally finite refinement.


Let $\UU$ be an open cover of $T$.


Let $\VV$ be an open $\sigma$-locally finite refinement of $\UU$ by hypothesis.


From Sigma-Locally Finite Cover has Locally Finite Refinement:

there exists a locally finite refinement $\AA$ of $\VV$


From Refinement of a Refinement is Refinement of Cover:

$\AA$ is a locally finite refinement of $\UU$


Since $\UU$ was arbitrary, it follows that:

every open cover of $T$ has a locally finite refinement

$\Box$


Axiom of Choice

This theorem depends on the Axiom of Choice.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.


Sources

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