Is a paracompact perfectly normal space hereditarily paracompact?

Question

Is the following implication true for topological spaces?

paracompact + perfectly normal $\implies$ hereditarily paracompact.

A proof or references would be appreciated.

Let $X$ be a topological space.

• $X$ is called paracompact if every open cover has a locally finite open refinement. No Hausdorff assumption here.
• $X$ is called perfectly normal if every closed set is a zero set; equivalently, if $X$ is normal and every closed set is a $G_\delta$-set.
• $X$ is called hereditarily paracompact if every subspace is paracompact; equivalently, if every open subspace is paracompact.

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https://math.stackexchange.com/a/4915864 states in passing that a space which is both perfectly normal and paracompact is hereditarily paracompact.

An idea: use the Kolmogorov quotient to simplify the hypotheses and assume Hausdorff?

Answer 1

Let $X$ be a perfectly normal paracompact space. Note that this implies fully normal, since fully normal is equivalent to normal and paracompact. In particular any open cover of $X$ has a partition of unity subordinate to it (here I don't require partitions of unity to be locally finite; note how locally finite condition doesn't matter, see Paracompactness and partitions of unity by Mather). This property is more well-known for paracompact Hausdorff spaces, which are fully normal. In this setting we could also take Kolmogorov quotient to come to the same conclusion, because perfectly normal $T_0$ spaces are $T_6$.

For a function $s:X\to\mathbb{R}$ denote $\text{coz}(s) = \{x\in X : s(x) > 0\}$ to be the cozero set of $s$.

Let $Y\subseteq X$ and let $\mathcal{U} = (U_i)_{i\in I}$ be an open cover of $Y$ (here I mean for $U_i$ to be sets open in $X$). Write $X\setminus \bigcup \mathcal{U} = \bigcap_n V_n$ where $V_n$ are open. For each $n$, the family $\mathcal{U}_n = \{V_n\}\cup \mathcal{U}$ is an open cover of $X$. For each $n$ find a partition of unity $\{f_{i, n} : i\in I\}\cup \{g_n\}$ subordinate to $\mathcal{U}_n$ so that $\text{coz}(f_{i, n})\subseteq U_i$ and $\text{coz}(g_n)\subseteq V_n$. Consider $f_i = \sum_n 2^{-n} f_{i, n}$ and $g = \sum_n 2^{-n} g_n$. Both functions $f_i$ and $g$ are well-defined and continuous because the functions involved take values in $[0, 1]$, and are continuous. Moreover $$\sum_i f_i + g = \sum_n 2^{-n}\left(\sum_i f_i + g_n\right) = \sum_n 2^{-n} = 1.$$ If $y\in Y$ then $y\notin V_n$ for some $n$, and so $g_n(y) = 0$. This implies $1-g(y) \geq 2^{-n} > 0$. Consider $h_i:Y\to [0, 1]$ defined as $h_i(y) = \frac{f_i(y)}{1-g(y)}$. Then $h_i$ is well-defined, continuous, and $\sum_i h_i = 1$, so that $(h_i)$ is a partition of unity. Moreover $\text{coz}(h_i) = \text{coz}(f_i) = \bigcup_n \text{coz}(f_{i, n})\subseteq U_i$, and so $(h_i)$ is a partition of unity subordinate to $\mathcal{U}$. It follows that $Y$ is paracompact.

Answer 2

Here is a simple argument using results from [1].

Theorem 5.1.28. Paracompactness[3] is hereditary with respect to $F_\sigma$-sets.

Suppose $X$ is perfectly normal and paracompact, and $U$ is an open set in $X$. Because perfectly normality implies $G_\delta$-ness, $U$ is an $F_\sigma$ set in $X$, thus paracompact by the above theorem.

Therefore, every open subspace of $X$ is paracompact, i.e., $X$ is hereditarily compact [2].

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[1]: General Topology, Ryszard Engelking.

[2]: For every subspace $Y \subseteq X$, let $\mathcal O = \left\{ U_\alpha \cap Y \right\}_{\alpha \in I}$ be an open cover of $Y$ (where $U_\alpha$ is open in $X$), then $\bigcup_{\alpha \in I} U_\alpha$ is an open subset (subspace) of $X$, we can lift the cover to $\bigcup_{\alpha \in I} U_\alpha$ to get an locally finite open refinement.

[3] In [1], paracompactness refers to “$T_2$ and paracompact”. So, if $X$ is $T_0$, we can use “perfectly normal” to get $T_2$ and get the result. If $X$ is not $T_0$, we can pass the Kolmogorov quotient because all the notion in the proof is invariant under Kolmogorov quotient.

Answer 3

A space is countably paracompact if every countable open cover has a locally-finite open refinement. A space is $\sigma$-paracompact if every open cover has a $\sigma$-locally-finite open refinement. Michael showed that a regular space is paracompact if and only it is $\sigma$-paracompact, but the following statement is not so well known.

Theorem ([Tall, Th.1.17]) A space is paracompact if and only if it is both countably paracompact and $\sigma$-paracompact. $\quad\blacksquare$

Since every perfectly normal space is countably paracompact, it follows that a perfectly normal space is paracompact if and only if it is $\sigma$-paracompact.

Proposition The property of being $\sigma$-paracompact is inherited by $F_\sigma$-subsets.

Proof This is an easy modification of the proof of [Eng, Th.5.1.28], which shows (without using regularity) that every $F_\sigma$-subset of a paracompact space is $\sigma$-paracompact. $\quad\blacksquare$

Proposition If every open subspace of a space $X$ is paracompact, then $X$ is hereditarily paracompact.

Proof This is not difficult and is given as Exercise 5.1.F in [Eng]. $\quad\blacksquare$

A space is perfect if each of its open subsets is an $F_\sigma$-set (equivalently, if each of its closed subsets is a $G_\delta$-set).

Proposition A space $X$ which is paracompact, perfect, and hereditary countably paracompact is hereditarily paracompact.

Proof It suffices that each open subset of $X$ is paracompact. As each open subset of $X$ is an $F_\sigma$-set, it is $\sigma$-paracompact. It is countably paracompact by assumption, hence paracompact. $\quad\blacksquare$

Corollary Every perfectly normal paracompact space is hereditarily paracompact.

Proof The property of being perfectly normal is hereditary and implies countable paracompactness [Eng, Co.5.2.5]. $\quad\blacksquare$

A more direct proof to this uses the fact that the property of being countably paracompact normal is inherited by $F_\sigma$-subsets.

References

[Eng] R. Engelking, General Topology, second edition, Heldermann Verlag Berlin, (1989).

[Tall] F. Tall, Set-theoretic Consistency Results and Topological Theorems Concerning the Normal Moore Space Conjecture and Related Problems, Polska Akademia Nauk, Inst. Mat, (1977)