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After reading that every perfectly normal space is countably paracompact, I wanted to find an example of a space that is $T_5$, but isn't countably paracompact. However, no examples of spaces verifying this can be found in $\pi$-Base yet.

I have found examples of spaces that are completely normal and $T_0$, but aren't countably paracompact. But I haven't found any spaces that are completely normal and $R_0$, but aren't countably paracompact. Could anyone give an example of a space verifiying the last conditions (completely normal + $R_0$ + $\neg$countably paracompact)?

Note: if it's possible, an example of a space verifying: ($T_5$ + $\neg$countably paracompact).

Lucenaposition
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Almanzoris
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1 Answers1

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In the article A Small Dowker Space in ZFC by Zoltan T. Balogh they prove that there exists a $T_5$ Dowker space of cardinality $\mathfrak{c}$.

Recall that a Dowker space is a $T_4$ space which isn't countably paracompact.

Note that you are looking for Dowker spaces, which gained notoriety of being tricky to give examples of.

Jakobian
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