Example of a compact set $K$ and a cover of it by closed sets such that the cover contains no countable subcover of $K$.
So my $K$ would be $[0,1]$ and my cover would be $[0,\frac{1}{2}]\cup[\frac{1}{2},1]$. Would that work?
2026-04-19 02:00:12.1776564012
uncountable subcovers of a compact set
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Nope! Because there are 2 items in your cover and that is a countable sub cover.
As every singleton set is closed we could consider our cover to be the union of all singleton set for real numbers in $[0,1]$. As the interval $[0,1]$ is not countable over the reals and removing any one of the singleton sets makes it no longer a cover this satisfies the conditions.