Contents

# Contents

## Definition

Recall:

###### Definition

(locally finite cover)

Let $(X,\tau)$ be a topological space.

An open cover $\{U_i \subset X\}_{i \in I}$ of $X$ is called locally finite if for each point $x \in X$, there exists a neighbourhood $U_x \supset \{x\}$ such that it intersects only finitely many elements of the cover, hence such that $U_x \cap U_i \neq \emptyset$ for only a finite number of $i \in I$.

###### Definition

(refinement of open covers)

Let $(X,\tau)$ be a topological space, and let $\{U_i \subset X\}_{i \in I}$ be a open cover.

Then a refinement of this open cover is a set of open subsets $\{V_j \subset X\}_{j \in J}$ which is still an open cover in itself and such that for each $j \in J$ there exists an $i \in I$ with $V_j \subset U_i$.

Now:

###### Definition

(paracompact topological space)

A topological space $(X,\tau)$ is called paracompact if every open cover of $X$ has a refinement (def. ) by a locally finite open cover (def. ).

###### Remark

(differing terminology)

As with the concept of compact topological spaces (this remark), some authors demand a paracompact space to also be a Hausdorff topological space. See at paracompact Hausdorff space.

## Examples

###### Example

Every compact space is paracompact.

###### Example

Every locally connected locally compact topological group is paracompact (this prop.).

###### Proposition

locally compact and second-countable Hausdorff space are paracompact.

###### Example

(Euclidean space is paracompact)

For $n \in \mathbb{N}$, then the Euclidean space $\mathbb{R}^n$, regarded with its metric topology is paracompact.

###### Proof

Euclidean space is evidently locally compact and sigma-compact. Therefore the statement follows since locally compact and sigma-compact spaces are paracompact (prop. ).

###### Proposition

Paracompactness is preserved by forming disjoint union spaces (coproducts in Top).

###### Proof

Consider a disjoint union $X = \coprod X_\lambda$ whose components are paracompact. As the union is disjoint, the components, that is to say, the $X_\lambda$, are open in $X$. Thus any open cover, say $\mathcal{U}$, of $X$ has a refinement by open sets, say $\mathcal{V}$, such that each $V \in \mathcal{V}$ is contained in some $X_\lambda$. Thus we can write $\mathcal{V} = \coprod \mathcal{V}_\lambda$. As each $X_\lambda$ is paracompact, each $\mathcal{V}_\lambda$ has a locally finite refinement, say $\mathcal{W}_\lambda$. Then let $\mathcal{W} := \coprod \mathcal{W}_\lambda$. As each $\mathcal{W}_\lambda$ is a refinement of the corresponding $\mathcal{V}_\lambda$, $\mathcal{W}$ is a refinement of $\mathcal{V}$, and hence of $\mathcal{U}$. As each point of $X$ has a neighbourhood which meets only elements of one of the $\mathcal{W}_\lambda$, and as that $\mathcal{W}_\lambda$ is locally finite, $\mathcal{W}$ is locally finite. Thus $\mathcal{U}$ has a locally finite refinement.

## Properties

### General

###### Lemma

Let $X$ be a paracompact Hausdorff space, and let $\{U_i \subset X\}_{i \in I}$ be an open cover. Then there exists a countable cover

$\{V_n \subset X\}_{n \in \mathbb{N}}$

such that each element $V_n$ is a union of open subsets of $X$ each of which is contained in at least one of the elements $U_i$ of the original cover.

(e.g. Hatcher, lemma 1.21)

###### Proof

Let $\{f_i \colon X \to [0,1]\}_{i \in I}$ be a partition of unity subordinate to the original cover, which exists since paracompact Hausdorff spaces equivalently admit subordinate partitions of unity.

For $J \subset I$ a finite set, let

$V_J \;\coloneqq\; \left\{ x \in X \;\vert\; \underset{j \in J}{\forall} \left( \underset{k \in I \setminus J}{\forall} \left( f_j(x) \gt f_k(x) \right) \right) \right\} \,.$

By local finiteness there are only a finite number of $f_k(x)$ greater than zero, hence the condition on the right is a finite number of strict inequalities. Since the $f_i$ are continuous, this implies that $V_J$ is an open subset.

Moreover, $V_J$ is contained in $supp(f_j)$ for $j \in J$ and hence in one of the $U_i$.

Now for $n \in \mathbb{N}$ take

$V_n \;\coloneqq\; \underset{ {J \subset I} \atop { {\vert J\vert} = n } }{\cup} V_J$

to be the union of the $V_J$ over all subset $J$ with precisely $n$ elements.

The set $\{V_n \subset X\}_{n \in \mathbb{N}}$ is a cover because for any $x \in X$ we have $x \in V_{J_x}$ for

$J_x \coloneqq \{ i \in I \;\vert\; f_i(x) \gt 0 \}$

(which is finite by local finitness of the partition of unity).

### Homotopy and Cohomology

###### Proposition

On a paracompact space $X$, every hypercover of finite height is refined by the Čech nerve of an ordinary open cover.

For a hypercover of height $n \in \mathbb{N}$, this follows by intersecting the open covers that are produced by the following lemma for $0 \leq k \leq n$.

###### Lemma

For $X$ a paracompact topological space, let $\{U_\alpha\}_{\alpha \in A}$ be an open cover, and let each $(k+1)$-fold intersection $U_{\alpha_0, \cdots, \alpha_{k}}$ be equipped itself with an open cover $\{V^{\alpha_0, \cdots, \alpha_k}_{\beta}\}$.

Then there exists a refinement $\{U'_{\alpha'}\}$ of the original cover, such that each $(k+1)$-fold intersection $U'_{\alpha'_0, \cdots, \alpha'_k}$ for all indices distinct is contained in one of the $V_\beta$.

This appears as (HTT, lemma 7.2.3.5).

The notion of paracompact space was introduced in

• Jean Dieudonné, Une généralisation des espaces compacts, Journal de Mathématiques Pures et Appliquées, Neuvième Série, 23: 65–76 (1944)

That fully normal spaces are equivalently paracompact is due to

• A. H. Stone, Paracompactness and product spaces, Bull. Amer. Math. Soc. Volume 54, Number 10 (1948), 977-982. (Euclid)

General accounts include

A basic discussion with an eye towards abelian sheaf cohomology and abelian Čech cohomology is around page 32 of

• Rudolf Fritsch, Renzo A. Piccinini, Cellular structures in topology, Cambridge studies in advanced mathematics Vol. 19, Cambridge University Press (1990). (pdf)

Discussion of paracompactness of CW-complexes includes

• Hiroshi Miyazaki, The paracompactness of CW-complexes, Tohoku Math. J. (2) Volume 4, Number 3 (1952), 309-313. 1952 Euclid