Proposition 7

Let $X$ be a set, $(Z_i)_{i \in I}$ a family of topological spaces, $(J_\lambda)_{\lambda \in L}$ a partition of $I$, and $(Y_\lambda)_{\lambda \in L}$ a family of sets indexed by $L$. Also, for each $\lambda \in L$ let $h_\lambda$ be a mapping of $Y_\lambda$ into $X$; for each $\lambda \in L$ and each $i \in J_\lambda$ let $g_{\lambda i}$ be a mapping of $Z_i$ into $Y_\lambda$, and put $f_i = h_\lambda \circ g_{\lambda i}$. If each $Y_\lambda$ carries the finest topology for which the mappings $g_{\lambda i} (i \in J_\lambda)$ are continuous, then the finest topology on $X$ for which the $f_i$ are continuous is the same as the finest topology for which the $h_\lambda$ are continuous.

Examples

1) Quotient topology. Let $X$ be a topological space, $R$ an equivalence relation on $X$, $Y = X/R$ the quotient set of $X$ with respect to the relation $R$, $\varphi : X \to Y$ the canonical mapping. The finest topology on $Y$ for which $\varphi$ is continuous is called the quotient of the topology of $X$ by the relation $R$; we shall study it in more detail in § 3.

2) Greatest lower bound of a set of topologies. Every family $(\mathfrak{T}_i)_{i \in I}$ of topologies on a set $X$ has a greatest lower bound $\mathfrak{T}$ in the set of all topologies on $X$, i.e. $\mathfrak{T}$ is the finest of all the topologies on $X$ which are coarser than each of the $\mathfrak{T}_i$. To see this we may apply Proposition 6, taking $Y_i$ to be the set $X$ with the topology $\mathfrak{T}_i$, and $f_i$ to be the identity mapping $Y_i \to X$. If $\mathscr{D}_i$ is the set of subsets of $X$ which are open in the topology $\mathfrak{T}_i$, then the set $\bigcap_{i \in I} \mathscr{D}_i$ is the set of subsets of $X$ which are open in $\mathfrak{T}$. $\mathfrak{T}$ is also called the intersection of the topologies $\mathfrak{T}_i$.

3) Sum of topological spaces. Let $(X_i)_{i \in I}$ be a family of topological spaces, $X$ the set which is the sum of the $X_i$ (Set Theory, Chapter II, § 4, no. 8, Definition 8); for each $i \in I$, let $j_i$ be the canonical (injective) mapping of $X_i$ into $X$. The finest topology $\mathfrak{T}$ on $X$ for which the mappings $j_i$ are continuous is called the sum of the topologies of the $X_i$, and $X$ with this topology is said to be the sum of the topological spaces $X_i$. Let us identify each of the $X_i$ with a subset of $X$ by means of $j_i$; then a set $A \subset X$ is open (resp. closed) in the topology $\mathfrak{T}$ if and only if each of the sets $A \cap X_i$ is open (resp. closed) in $X_i$. In particular, each of the $X_i$ is both open and closed.

The following proposition generalizes the situation of Example 3: