Definition of Connected Spaces

Topology · Axiom Academy

LESSON Definition of Connected Spaces Understanding when a topological space cannot be separated into disjoint parts Step 1: Intuition - What Does "Connected" Mean? Before we dive into the formal definition, let's build some intuition. In everyday language, we say something is "connected" if it's all in one piece. A rope is connected, but if you cut it, it becomes disconnected. In topology, we want to capture this same idea: a space is connected if it cannot be split into two separate, non-empty pieces that don't touch each other. A topological space is called connected if it cannot be written as the union of two disjoint, non-empty, open sets. Equivalently, is connected if there do not exist non-empty open sets such that: Let's unpack this definition carefully: Open sets: We use open sets because they're the fundamental building blocks in topology Disjoint: The sets and have no points in common Union equals X: Together, and make up the entire space Non-empty: Both sets must contain at least one point (otherwise the splitting is trivial) Step 3: What About Disconnected Spaces? A space is disconnected if it IS possible to write it as such a union. In this case, we call and a separation or disconnection of the space. A topological space is disconnected if there exist non-empty open sets such that , , and . Example: Two Separated Intervals Consider the space with the subspace topology from . This space is disconnected because we can write:

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