Let $X$ be a complete metric space. $E$ is a non empty open set in $X$ Then is $E$ a first category set or second category set.

Question

Let $X$ be a complete metric space. $E$ is a non-empty open subset of $X$

Then is $E$ a first category set or a second category set. it seems to me that $E$ should be a second category set, but how to prove that .

or my guess is wrong ? Any hint

Answer 1

$E$ is completely metrisable too, so it's a second category set.

Direct sketch of proof for the former: If $d$ is a complete metric on $X$ then $$d_1(x,y):=d(x,y)+ \left|\frac{1}{d(x,X\setminus E)} - \frac{1}{d(y,X\setminus E)}\right|$$

is a complete metric for $E$ (in its subspace topology), or use a general theorem that a $G_\delta$ subset of a completely metrisable space is completely metrisable, if your text covers that fact.

Added: this not the whole story yet: we need an extra argument to show it is second category in $X$ too, using the openness in a different way:

If $N$ is nowhere dense in $X$ and $N \subseteq E$, then $N$ is nowhere dense in $E$ too. If not, $\overline{N}^{(E)} = \overline{N}^{(X)} \cap E$ would have non-empty interior, and as $E$ is open this cannot happen.

It follows that $E$ is not second category in $X$ because it would be second category in itself, which we saw above it is not.