Connected Subsets

Topology · Axiom Academy

Understanding how subsets inherit and preserve connectedness Step 1: Connected Subsets in the Subspace Topology When we have a subset of a topological space , we need to understand what it means for to be connected. A subset of a topological space is called connected if is connected as a topological space with the subspace topology. Equivalently, is connected if it cannot be written as where are disjoint, non-empty, relatively open sets in . Step 2: Characterization Using the Original Topology We can check if a subset is connected without explicitly working with the subspace topology. A subset of is connected if and only if there do not exist open sets in such that: This theorem allows us to test connectedness by looking at how open sets in the larger space intersect with our subset. Step 3: Unions of Connected Sets When do unions of connected sets remain connected? The answer depends on whether the sets have common points. Let be a family of connected subsets of . If , then is connected. Proof Idea: Any separation of the union would force the common point into one of the parts, pulling all the connected sets into that same part, contradicting the separation. Step 4: When Unions Fail to Be Connected Let's see a concrete example where the union of connected sets is disconnected. In with the standard topology, consider: (a closed interval - connected) Then and , but is disconnected . We can separate this union as where both parts are open in the subspace topology of .

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