Continuous Maps on Topological Spaces Review

# Continuous Maps on Topological Spaces Review

We will now review some of the recent material regarding continuous maps on topological spaces.

Let $X$ and $Y$ be topological spaces and let $f : X \to Y$.

• Recall from the Continuous Maps on Topological Spaces page that $f$ is said to be Continuous at a Point $a \in X$ if there exists local bases $\mathcal B_a$ of $a$ and $\mathcal B_{f(a)}$ of $f(a)$ such that for all $B \in \mathcal B_{f(a)}$ there exists a $B' \in \mathcal B_a$ such that:
(1)
\begin{align} \quad f(B') \subseteq B \end{align}
• Furthermore, $f$ is said to be Continuous or more specifically, Continuous on All of $X$ if $f$ is continuous at every $a \in X$.
(2)
\begin{align} \quad f(U) \subseteq V \end{align}
• We then showed that if for all open sets $V$ in $Y$ we have that the inverse image $f^{-1}(V)$ is open in $X$ then this implies that for every basis $\mathcal B_Y$ of $Y$ and for every $B \in \mathcal B_Y$ we have that $f^{-1}(B)$ is open in $X$ (since every element in the basis of the topology on $Y$ is open by definition).
• Furthermore, we showed that if for every basis $\mathcal B_Y$ of $Y$ we have that for every $B \in \mathcal B_Y$ we have that $f^{-1}(B)$ is open in $X$ then this implies that $f$ is continuous on all of $X$ which completes our cycle of equivalent statements.
• On the Continuity of the Composition of Continuous Maps on Topological Spaces page we saw that if $X$, $Y$, and $Z$ were all topological spaces and $f : X \to Y$ and $g : Y \to Z$ are continuous functions then the composite function $g \circ f : X \to Z$ defined for all $x \in X$ by $(g \circ f)(x) = g(f(x))$ is also inherently continuous on all of $X$.