Example 2

Let X be a topological space, let B be a topological space and let A be a subspace of B, and let $f: A\rightarrow X$ be a continuous mapping inducing a homeomorphism of A onto its image.

Put $A'=f(A)$; let $f'$ be the mapping of $A'$ into B which associates to each $x\in A'$ the unique inverse image of $x$ by $f$. The mapping $f'\circ f$ is continuous; since $f$ is strict, the mapping $f'$ is continuous. There exists a unique continuous mapping $u$ of the space $X\cup_fB$ into the space $B\cup_{f'}X$ which maps $\alpha_f(x)$ to $\beta_{f'}(x)$ for $x\in X$ and $\beta_f(b)$ to $\alpha_{f'}(b)$ for $b\in B$; it is a homeomorphism mapping the subspace $\alpha_f(X)$ onto the subspace $\beta_{f'}(X)$.