Properties of Continuous Maps

Topology · Axiom Academy

Understanding continuity in topological spaces and its fundamental properties Step 1: The Definition of Continuity In topology, continuity is defined quite differently from calculus. Instead of epsilon-delta arguments, we use the language of open sets. This elegant definition captures the essence of continuity: continuous functions preserve the structure of open sets . Let and be topological spaces. A function is continuous if for every open set in , the preimage is open in . Step 2: Composition of Continuous Maps One of the most important properties of continuous functions is that they compose well. This makes topology much more structured and workable. Theorem: Composition of Continuous Maps If and are continuous maps, then the composition is also continuous. Why is this true? Let's think through the proof: Since is continuous, is open in Step 3: Restriction to Subspaces When we restrict a continuous function to a subspace, we don't lose continuity. This is extremely useful for studying local properties of functions. Theorem: Restriction is Continuous Let be continuous and let be a subspace of . Then the restriction is continuous. Intuition: If is continuous, it respects the open set structure of the whole space. When we restrict to a subspace , we're just looking at a "smaller version" of the same structure, so continuity is preserved. Step 4: Continuous Images of Compact Sets

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