Path Connected vs Connected

Topology · Axiom Academy

LESSON Path Connected vs Connected Understanding the subtle relationship between two fundamental topological properties Step 1: Recall the Definitions In topology, connectedness and path connectedness are two ways of describing whether a space "comes in one piece." Let's recall their formal definitions. A topological space is connected if it cannot be written as the disjoint union of two non-empty open sets. Equivalently, is connected if the only subsets that are both open and closed are and itself. A topological space is path-connected if for any two points , there exists a continuous function (called a path ) such that and . Step 2: Path-Connected Implies Connected Let's establish our first major result: every path-connected space is connected. This is an important theorem in topology. If is path-connected, then is connected. Suppose is path-connected but not connected. Then can be written as where and are non-empty, open, and disjoint. Since is path-connected, there exists a continuous path with and . Consider the sets and . These are open in (by continuity of ), non-empty, disjoint, and their union is . This means is disconnected. But the interval is connected in (a fundamental theorem of real analysis). Contradiction! Therefore, must be connected. □ Step 3: Connected Does NOT Imply Path-Connected You might wonder: does the converse hold? If a space is connected, must it be path-connected?

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