A closed set in $(A, \mathcal T_A)$ that is not closed in $(X, \mathcal T)$

Question

Give an example of a topological space $(X, \mathcal T)$, a subspace $(A, \mathcal T_A)$ of $(X, \mathcal T)$, and a closed set in $(A, \mathcal T_A)$ that is not closed in $(X, \mathcal T)$.

All the sets I create are closed in both and I am stuck.

Answer 1

Let $(X,T)$ be $\mathbb{R}$ with the standard topology, and let $A = (0,1)$ with the subspace topology. Then $(0,1/2]$ is closed in $A$ but not in $X$.

Answer 2

Let $X= \mathbb R$ with $\tau$ as the Euclidean topology. Pick an open subset $A$ of $\mathbb R$, and endow it with the relative topology $\tau_A$.

It should be easy to see that $A$ is closed in $\tau_A$ but $A^c \notin \tau_A$.

Answer 3

Try $X=\mathbb{R}^2$ and let $A$ be the open unit disk.