If $fmM:X\to Y$ is continuous then $fmM:X\to f(X)$ is continuous

Theorem 1.

Suppose $X\mathrm{,}Y$ are topological spaces and $f\mathrm{:}X\mathrm{\to }Y$ is acontinuous function. Then$f\mathrm{:}X\mathrm{\to }f\mathit{}\mathrm{(}X\mathrm{)}$ is continuous when $f\mathit{}\mathrm{(}X\mathrm{)}$ is equipped withthe subspace topology.

Proof.

Let us first note that using aproperty on this page (http://planetmath.org/InverseImage), we have

$$X=f^{-1}f(X).$$

For the proof, suppose that $A$ is openin $f(X)$, that is, $A=U\cap f(X)$for some open set $U\subset Y$. From the properties ofthe inverse image, we have

$$f^{-1}(A)=f^{-1}(U)\cap f^{-1}(f(X))=f^{-1}(U)$$

so $f^{-1}(A)$ is open in $X$.∎