Continuous Functions in Topology

Topology · Axiom Academy

Understanding continuity through the lens of open sets Step 1: The Topological Definition of Continuity In calculus, we learned about continuous functions using limits and epsilon-delta arguments. In topology, we have a completely different—and more powerful—definition based on open sets. Let be topological spaces. A function is continuous if and only if for every open set in , the preimage is open in . In other words: continuous functions are exactly those that pull back open sets to open sets. Visualization: A continuous function f maps an open set in X to create an open preimage Before we go further, let's make sure we understand what a preimage is. Given a function and a subset , the preimage is: This is the set of all points in that map into . Points in the preimage (left) map into the target set V (right) Step 3: The Closed Set Characterization There's an equivalent way to characterize continuous functions using closed sets instead of open sets: A function is continuous if and only if for every closed set in , the preimage is closed in . Why is this equivalent? Remember that a set is closed if and only if its complement is open. So: If is closed in , then is open in But this equals , so is closed! Step 4: Connection to the ε-δ Definition If you've taken calculus, you know the ε-δ definition of continuity. For functions , these two definitions are equivalent! A function is continuous at if for every , there exists such that:

This is the written version of the interactive lesson above. See the full Topology course.