Proposition 4.3.1.
If \(f:A\to V\) is a continuous function and \(U\subset V\) is an open set, then
\begin{equation*}
f^*(U)=\left\{x\in A\ \middle\vert\ f(x)\in U\right\}
\end{equation*}
is an open set.
Section 4.3 continuity and sets
Continuous functions will interact dramatically with various topological notions, and that will end up proving some really handy theorems almost automatically for us.
Proposition 4.3.2. converse of Proposition 4.3.1.
A function \(f:A\to V\) with the property that, for any open set \(U\subseteq V\text{,}\) \(f^{*}(U)\) is open is continuous.
Taken together, Proposition 4.3.1 and Proposition 4.3.2 are called the topological characterization of continuity.
Remark 4.3.3.
In Proposition 4.3.1 and Proposition 4.3.2, I’ve used the notation \(f^*(U)\) to denote the preimage of the set \(U\text{.}\) Many texts will use the notation \(f^{-1}(U)\) for this, but that’s terrible for all sorts of reasons. I’ll let you try to figure out what they are.
Checkpoint 4.3.4.
Prove the Closed-Set Characterization of Continuity:
\(f:V\to W\) is continuous if and only if for every closed \(K\subseteq W\text{,}\) \(f^*(K)\subseteq V\) is closed.
Remark 4.3.5.
Modern mathematics is built from sets with structure and structure-preserving maps. For example, in linear algebra one studies vector spaces (a set with additional structure) and linear transformations, which are functions that preserve the vector space structure. In group theory, we study groups and group homomorphisms.
Continuous functions preserve the analytic structure we’re interested in. We’ve seen several ways in which this general statement is true:
-
continuous functions preserve sequence limits (the Sequential Characterization of Continuity)
-
continuous functions allow us to translate distances in one metric to distances in another (Proposition 4.2.10)
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If \(f:V\to W\) is continuous, then the induced map \(f^*:2^W\to 2^V\) preserves openness
