If we have two topologies on $X$, such that $\mathcal{T} \subset \mathcal{K}$, and we have $A \subset X$ are the subspaces subsets of each other?

Question

Let's say we have two topologies on a set $X$, namely $\mathcal{T}$ and $\mathcal{K}$ such that $\mathcal{T} \subset \mathcal{K}$.

Given a subset $A \subset X$, and letting $\mathcal{T}_A$ and $\mathcal{K}_A$ denote the respective subspace topologies, do we still have $\mathcal{T}_A \subset\mathcal{K}_A$?

Answer 1

Since $\mathcal{T} \subset \mathcal{K}$, and given $A \subset X$, we have the following subspace topologies.

$\mathcal{T}_A = \{ A \cap U \ | \ U \in \mathcal{T}\}$ and $\mathcal{K}_A = \{ A \cap U \ | \ U \in \mathcal{K}\}$

Since $U \in \mathcal{T} \implies U \in \mathcal{K}$ we have $ A \cap U \in \mathcal{T}_A \implies A \cap U \in \mathcal{K}$ so that $\mathcal{T}_A \subset \mathcal{K}_A$. $\ \square$