Restriction of a Continuous Map, and Continuous Images of Compact Subsets

Let $(X,\mathcal{T}_{X})$ and $(Y,\mathcal{T}_{Y})$ be \reftext{def:topological-space-2026a}{topological spaces}, let $f:X\to Y$ be a \reftext{def:continuous-map-topological-spaces-2026a}{continuous map}, and let $A\subseteq X$ be equipped with the \reftext{def:subspace-topology-2026a}{subspace topology} $\mathcal{T}_{A}$. Let $f|_{A}:A\to Y$ denote the map with $f|_{A}(x)=f(x)$ for every $x\in A$, and write $f(A)=\{f(x):x\in A\}$. Then the following hold.

\textbf{1. (Restriction)} The map $f|_{A}$ is continuous from $(A,\mathcal{T}_{A})$ to $(Y,\mathcal{T}_{Y})$.

\textbf{2. (Image of a compact subset)} If $A$ is \reftext{def:compact-space-and-subset-2026a}{compact in $X$}, then $f(A)$ is compact in $Y$.