Definition 10

The quotient space of X by R is denoted by $X/A$ and is called the space obtained from X by contraction of the subset A.

Let us denote by $\rho : X\rightarrow X/A$ the canonical surjection. Let Y be a topological space and let $f: X\rightarrow Y$ be a continuous mapping. In order that there should exist a continuous mapping $g: X/A\rightarrow Y$ such that $g\circ \rho =f$, it is necessary and sufficient that $f$ should be constant on A.

If the set A is closed (resp. open) in X, the mapping $\rho$ is closed (resp. open) and induces a homeomorphism of X-A onto its image. If A is a closed and quasi-compact subset of X, the mapping $\rho$ is proper (TG, I, p. 75, th. 1).

If the set A is empty, $\rho$ is a homeomorphism. If it is not empty, A is a point of $X/A$ called the base point of $X/A$ and denoted by $s_{X/A}$. The space $X/A$ is then identified with the space obtained by attaching the space X to the space $\{s_{X/A}\}$ by means of the unique mapping from A into $\{s_{X/A}\}$.