Remark

Let A be a topological space, let B be a closed subset of A, and let U be an open neighbourhood of B in A. Suppose that the set U-B is the union of a finite family $(M_{\ell})_{\ell\in L}$ of pairwise disjoint open sets. For each $\ell \in L$, put $N'_{\ell}= M_{\ell}\cup B$. Let $C'$ denote the topological sum of the spaces A-B and $N'_{\ell}$, for $\ell \in L$; let also $\varphi : A-B\rightarrow C'$ and $\varphi_{\ell}: N'_{\ell}\rightarrow C'$, for $\ell \in L$, denote the canonical injections. Let C be the quotient topological space of $C'$ by the finest equivalence relation S for which the points $\varphi (x)$ and $\varphi_{\ell}(x)$ are equivalent, for each $\ell \in L$ and each $x\in M_{\ell}$; let $\rho : C'\rightarrow C$ be the canonical mapping.

Let us prove that the space A is recovered from the space C by identification of the sets $B_{\ell}$ by means of the homeomorphisms $\rho \circ$ $(\varphi_{\ell}|B): B\rightarrow B_{\ell}$.

For every $\ell \in L$, the set $M_{\ell}$ is open in A-B and in $N'_{\ell}$. The mappings $\rho \circ \varphi$ and $\rho \circ \varphi_{\ell}$ are homeomorphisms of the spaces A-B and $N'_{\ell}$, for $\ell \in L$, onto open subsets of C (TG, I, p. 17, Prop. 9). For every $\ell \in L$, the set $\varphi_{\ell}(B)$ is closed in $C'$ and saturated for the equivalence relation S. Hence the set $B_{\ell}=\rho (\varphi_{\ell}(B))$ is closed in C and the mapping $\rho \circ \varphi_{\ell}$ induces a homeomorphism of B onto $B_{\ell}($loc. cit.). Analogously, for every $\ell \in L$, the set $N_{\ell}=\rho (\varphi_{\ell}(N'_{\ell}))$ is an open neighbourhood of $B_{\ell}$; the sets $N_{\ell}$ are mutually disjoint.

By passing to the quotient, the mapping $f': C'\rightarrow A$ deduced from the canonical injections into A of the spaces A-B and $N_{\ell}$, for $\ell \in L$, induces a

continuous mapping $f: C\rightarrow A$. If $x$ is a point of B, the fibre $\overset{-1}{f}(x)$ is the set of points $\rho (\varphi_{\ell}(x))$, for $\ell \in L$. The equivalence relation on C associated with the mapping $f$ is the relation R defined at the beginning of the number. Let $B_L$ denote the union of the family $(B_{\ell})_{\ell\in L}$. By construction, the mapping $f$ induces a homeomorphism of $C-B_L$ onto A-B and, for $\ell \in L$, a homeomorphism of $N'_{\ell}$ onto $N_{\ell}$. We shall prove that the mapping $f$ is closed; the topology of A will therefore be the quotient topology of C by the equivalence relation R (I, p. 18, Example 2). For this, let us prove that if F is a subset of A such that $F\cap (A-B)$ is closed in A-B and such that $F\cap N'_{\ell}$ is closed in $N'_{\ell}$, for $\ell \in L$, the set F is closed in A. The set U, union of the family $(N'_{\ell})_{\ell\in L}$, is open in A and the sets $N'_{\ell}$, for $\ell \in L$, constitute a finite closed covering of it. Hence the set $F\cap U$ is closed in U (TG, I, p. 18, Prop. 3). The sets U and A-B form an open covering of A, therefore the set F is closed in A (loc. cit.).