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added LaTeX formatting and edited title a bit.
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Marco Golla
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Is an open subset of X paracompact when X is a compact subset of a Hausdorff locally convex tvsTVS paracompact?

This repeats the title in a more readable way. Take a compact subset X$X$ of a Hausdorff locally convex topological vector space and U$U$ be an open subset of X$X$. Is U$U$ paracompact?

Is an open subset of X paracompact when X is a compact subset of a Hausdorff locally convex tvs?

This repeats the title in a more readable way. Take a compact subset X of a Hausdorff locally convex topological vector space and U be an open subset of X. Is U paracompact?

Is an open subset of a compact subset of a Hausdorff locally convex TVS paracompact?

This repeats the title in a more readable way. Take a compact subset $X$ of a Hausdorff locally convex topological vector space and $U$ be an open subset of $X$. Is $U$ paracompact?

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Is an open subset of X paracompact when X is a compact subset of a Hausdorff locally convex tvs?

This repeats the title in a more readable way. Take a compact subset X of a Hausdorff locally convex topological vector space and U be an open subset of X. Is U paracompact?