Continuous Map

Let $(X, \tau_X)$ and $(Y, \tau_Y)$ be topological spaces.

Definition. A function $f : X \to Y$ is a continuous map if for every open set $V \in \tau_Y$, the preimage $f^{-1}(V) \in \tau_X$.

Proposition. The following are equivalent:

1. $f$ is continuous.
2. The preimage of every closed set in $Y$ is closed in $X$.
3. For every $A \subseteq X$, $f(\mathrm{cl}(A)) \subseteq \mathrm{cl}(f(A))$.
4. For every $x \in X$ and every open set $V$ containing $f(x)$, there exists an open set $U$ containing $x$ with $f(U) \subseteq V$.

Proof sketch. (1)$\Leftrightarrow$(2): complements commute with preimages. (1)$\Rightarrow$(3): if $C \supseteq f(A)$ is closed, then $f^{-1}(C) \supseteq A$ is closed, so $\mathrm{cl}(A) \subseteq f^{-1}(C)$, giving $f(\mathrm{cl}(A)) \subseteq C$; take $C = \mathrm{cl}(f(A))$. (3)$\Rightarrow$(1): if $V$ is open, set $A = X \setminus f^{-1}(V)$; then (3) gives $\mathrm{cl}(A) = A$, so $f^{-1}(V)$ is open. (1)$\Leftrightarrow$(4) is immediate from the definition. $\square$

Proposition ($\varepsilon$-$\delta$ equivalence). If $(X, d_X)$ and $(Y, d_Y)$ are metric spaces with their metric topologies, then $f : X \to Y$ is continuous if and only if for every $x \in X$ and every $\varepsilon > 0$, there exists $\delta > 0$ such that $d_X(x, x') < \delta$ implies $d_Y(f(x), f(x')) < \varepsilon$.

Proposition. The composition of continuous maps is continuous. The identity map $\mathrm{id}_X : X \to X$ is continuous.

Proposition. Topological spaces and continuous maps form a category $\mathbf{Top}$, with homeomorphisms as isomorphisms.

Examples.

1. Constant maps. Every constant function $f : X \to Y$ is continuous.
2. Inclusion maps. If $A \subseteq X$ carries the subspace topology, the inclusion $A \hookrightarrow X$ is continuous.
3. Projections. The projection $\pi_i : \prod_j X_j \to X_i$ from a product space is continuous (by definition of the product topology).

Remark. A continuous map $f : X \to Y$ induces a geometric morphism between the sheaf topoi $\mathbf{Sh}(X)$ and $\mathbf{Sh}(Y)$: the direct image $f_*$ pushes sheaves forward, and its left adjoint $f^*$ pulls them back.