Nontrivial example of closed set relative to subspace but not with respect to original space

Question

I am interested in finding the example of the set which is close relative subspace but not in original space.
I know that $\mathbb Q $ is the subspace of $\mathbb R$ And $\mathbb Q$ is close relative to itself but not with respect to $\mathbb R$.
But I am interested in the nontrivial example not like above I had mentioned.
Any Help will be appreciated

Answer 1

This really depends on your notion of trivial. The interval $(0,1]$ is closed in $(0,2)$, for example. Another example would be the open unit ball in $\mathbb{R}^2$ considered on the subspace of the latter, $B_1(0,0) \cup B_1(2,0)$. Or for example, if you take $X = \bigcup_{i \in \mathbb{Z}} L_i$ with $L_i = (0,1) \times \{i\}$, then each $L_i$ is closed in $X$.

More generally, $F$ is closed in a subspace $Y$ of a space $X$ iff it is of the form $F = C \cap Y$ with $C$ closed in $X$. This gives a way of generating examples, as non-trivial as you can come up with.