Product of paracompact and compact spaces

Asked Jan 09 '22 at 17:08

Active Dec 12 '22 at 06:52

Viewed 42 times

“If $X$ is paracompact and $Y$ is compact, then $X \times Y $ is also paracompact.”

From this , if $X$ is a discrete space and $Y = \{0, 1\}$ with the topology $\{\emptyset, Y, \{0\}\}$, then the product space $X × Y$ is paracompact.

How I can show that $Y$ is compact? Is it enough to say "since $Y$ is finite then $Y$ is compact" and why $X$ is paracompact?

edited Dec 12 '22 at 06:52

Martin Sleziak

asked Jan 09 '22 at 17:08

Marwa Zeyad

$X$ is paracompact because you are assuming it. – José Carlos Santos Jan 09 '22 at 17:13
It is not hard to show $X$ is paracompact since singleton sets are open in $X$ (take a refinement of a cover $C$ to be the collection of all singletons taken from members of $C$). – David Mitra Jan 09 '22 at 17:23
how i can show that Y is compact ? – Marwa Zeyad Jan 09 '22 at 17:24
$Y$ is compact since there are only 3 open sets. – David Mitra Jan 09 '22 at 17:25
if we want to show a set is compact we must show that for any open cover of Y is has a finite subcover can you help me how i can use this to prove that Y is compact – Marwa Zeyad Jan 09 '22 at 17:36
Any open cover has at most 3 sets. The cover itself serves as the required finite subcover. – David Mitra Jan 09 '22 at 17:41
A finite space is always compact so the theorem applies with this $Y$. $X$ is paracompact trivially (use singletons) or by the theorem that all metric spaces are paracompact; take your pick. – Henno Brandsma Jan 10 '22 at 06:43

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