Connectedness Under Continuous Maps

Topology · Axiom Academy

LESSON Connectedness Under Continuous Maps Understanding how continuous functions preserve connectedness Step 1: The Fundamental Question One of the most powerful ideas in topology is that certain properties are preserved under continuous maps. Today we explore a beautiful theorem: the continuous image of a connected space is connected . The visualization above shows how a continuous function maps a connected space (an interval) to another connected space. Notice that no matter how the function transforms the space, it cannot create a "gap" or "separation." Let be a continuous function between topological spaces and . If is connected, then is connected. This theorem is powerful because it tells us that connectedness is a topological invariant under continuous maps. It's one of the properties that continuous functions respect and preserve. It explains why continuous functions on intervals have the Intermediate Value Property It provides a way to prove that certain spaces are NOT homeomorphic It's fundamental to understanding how topology behaves under transformations We'll prove this theorem by contradiction . The strategy is elegant: assume the image is disconnected, and show that this forces the original space to be disconnected as well—contradicting our hypothesis. A space is disconnected if there exist non-empty open sets and in such that: Assume for contradiction that is disconnected. Then there exist non-empty open sets and in such that:

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