Paracompactness is Preserved under Projections

Theorem

Let $I$ be an indexing set.

Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.

Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$.

Let $\pr_\alpha: \struct {S, \tau} \to \struct {S_\alpha, \tau_\alpha}$ be the projection on the $\alpha$ coordinate.

If $\struct {S, \tau}$ is paracompact, then each of $\struct {S_\alpha, \tau_\alpha}$ is also paracompact.

Proof

We are given that $\struct {S, \tau}$ is paracompact.

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

We are given that for each $\alpha \in I$, $\pr_{\alpha}: S \to S_{\alpha}$ is the $\alpha$-th projection.

For each $\alpha \in I$, let $\pr_{\alpha}^\gets: \powerset {S_{\alpha}} \to \powerset S$ be the inverse image mapping induced by $\pr_{\alpha}$

Let $T \subseteq S_\alpha$.

Let $\family {T_i}_{i \mathop \in I}$ be the family of sets defined by:

$T_i = \begin {cases} T & : i = \alpha \\ S_i & : i \ne \alpha \end {cases}$

By Inverse Image Mapping Induced by Projection, for all $\alpha \in I$, $\pr_\alpha^\gets$ is the mapping defined by:

$\ds \forall T \subseteq S_i: \map {\pr_{\alpha}^\gets} T = \prod_{i \mathop \in I} T_i$

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Therefore every open cover of $\struct {S_\alpha, \tau_\alpha}$ has an open refinement which is locally finite.

Hence every $\struct {S_\alpha, \tau_\alpha}$ is paracompact.

$\blacksquare$

Sources

• 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $3$: Compactness: Invariance Properties