An example of subset $A$ such that $A \cap K$ is open in $K$ for each compact set $K$, but $A$ is not open.
59194
Spaces in which "$A \cap K$ is closed for all compact $K$" implies "$A$ is closed."
136106
Is every open subset of $ \mathbb{R} $ uncountable?
90826
Constructing a subset not in $\mathcal{B}(\mathbb{R})$ explicitly
95993
Isn't every subset of a compact space compact?
10127
Example of a sequence of closed and connected sets whose intersection is disconnected
125962
The set $A'$ is closed in a $T_1$ space
73130
A simple question about open set
140570
Suppose $E\subseteq\mathbb{R}$ is closed and $K\subseteq\mathbb{R}$ is compact. Show that every open cover of $E\cap K$ has a finite subcover
157218
If $E\subseteq X$ is connected, $E\cap A\neq\emptyset$ and $E\cap(X\setminus A)\neq\emptyset$ then $E\cap\partial A\neq\emptyset$
79424
Don't understand proof that interior of a set is open
158498
$A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
74330
$\mathbb{R}^n$ not homeomorphic to open subset of $\mathbb{R}^m$
86309
Whether the set $A$ is a compact subset of $M_3(\Bbb R)$.
13415
How to show if $A \subseteq B$, then $A \cap B = A$?
127173
Closed subset of compact set is compact
81927
Example of closed and bounded in $\mathbb Q$ set that is not compact
117561
A set is infinite iff there is a one-to-one correspondent with one of its proper subsets?