Compactness in a Subspace Agrees with Compactness in the Ambient Space

Let $(X,\mathcal{T})$ be a topological space, let $B\subseteq X$, and let

$$\mathcal{T}_B=\{B\cap U : U\in\mathcal{T}\}$$

be the subspace topology on $B$, so that $(B,\mathcal{T}_B)$ is a topological space by The Subspace Topology is a Topology. Let $A\subseteq B$. Then the following hold.

1. (Transitivity of the subspace topology) The subspace topology on $A$ inherited from the topological space $(B,\mathcal{T}_B)$ and the subspace topology on $A$ inherited from $(X,\mathcal{T})$ are the same collection of subsets of $A$, that is,

$$\{A\cap V : V\in\mathcal{T}_B\}=\{A\cap U : U\in\mathcal{T}\}.$$

2. (Compactness does not depend on the ambient space) The subset $A$ is compact in $B$ if and only if $A$ is compact in $X$.