Example

Let $X$ be a topological space, $(X_i)_{i \in I}$ a covering of $X$, $Y$ the sum of the subspaces $X_i$ of $X$; then there is a partition $(Y_i)_{i \in I}$ of $Y$ into subspaces which are both open and closed, and for each $i \in I$ there is a homeomorphism $f_i : Y_i \to X_i$. Let $f : Y \to X$ be the continuous mapping which agrees with $f_i$ on $Y_i$ for each $i \in I$, and let $R$ be the equivalence relation $f(x) = f(y)$; the quotient space $Y/R$ is thus obtained by "pasting together" the $Y_i$ (§ 2, no. 5). Consider the bijection $g : Y/R \to X$ associated with $f$; in general $g$ is not a homeomorphism, as is shown by the example in which each $X_i$ consists of a single point and $X$ is not discrete. However, if the interiors of the $X_i$ cover $X$, or if $(X_i)$ is a locally finite closed covering of $X$, then $g$ is a homeomorphism: for if $U$ is any open subset of $Y$ which is saturated with respect to $R$, then for each $i \in I$ the set

is open in $X_i$, and the assertion follows from Proposition 3 of no. 1.

The following proposition gives a simple sufficient condition for $g$ to be a homeomorphism: