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I'm looking for an example of a space which is paracompact but not metrizable. The definition of paracompactness that I'm working with is that $(X,\tau)$ is paracompact if it is Hausdorff ($T_{2}$) and for every open cover there exists a locally finite open refinement.

I'd also like to know how well paracompactness is preserved in products. Thanks in advance.

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    Yeah,$I$did confuse it with $[0,\omega_{1})$. Thanks.2012-01-11

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The Sorgenfrey line is a classical example (besides the compact examples mentioned in the thread from the comments), also because its square (the Sorgenfrey plane) is not even normal, let alone paracompact, which shows that products of even 2 relatively nice paracompact spaces can fail to be paracompact.

As a positive result, the product of a paracompact and a compact Hausdorff space is paracompact.

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    Thanks Henno. The Sorgenfrey line seems like a good counter-example for many cases, more than I had suspected.2012-01-11
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$\pi$-Base is an online encyclopedia of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It lists the following paracompact, Hausdorff spaces that are not metrizable. You can learn more about any of them by visiting the search result.

Alexandroff Square

Appert Space

Arens-Fort Space

Closed Ordinal Space $[0,\Omega]$

Concentric Circles

Fortissimo Space

Helly Space

$I^I$

Lexicographic Ordering on the Unit Square

Radial Interval Topology

Right Half-Open Interval Topology

Single Ultrafilter Topology

Stone-Cech Compactification of the Integers

The Extended Long Line

Tychonoff Plank

Uncountable Fort Space

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    Yes, but sign up is free.2014-11-22