3. Continuity and Homeomorphisms

Continuity and Homeomorphisms

Definition of Continuity Between Topological Spaces

Continuity: A function \( f: X \rightarrow Y \) between two topological spaces \( (X, \tau_X) \) and \( (Y, \tau_Y) \) is said to be continuous if the preimage of every open set in \( Y \) is open in \( X \). That is, for every open set \( V \) in \( Y \), the set \( f^{-1}(V) \) is open in \( X \).

Homeomorphisms and Topological Equivalence

Homeomorphism: A function \( f: X \rightarrow Y \) between two topological spaces \( (X, \tau_X) \) and \( (Y, \tau_Y) \) is called a homeomorphism if it is bijective, continuous, and its inverse function \( f^{-1} \) is also continuous. In other words, \( f \) establishes a one-to-one correspondence between the points of \( X \) and \( Y \) while preserving the topological structure.

Topological Equivalence: Two topological spaces \( X \) and \( Y \) are said to be topologically equivalent if there exists a homeomorphism between them. In this case, the spaces are essentially the same from a topological perspective, even if their point sets may differ.

Properties Preserved Under Homeomorphisms

Homeomorphisms preserve various topological properties, including:

• Continuity of functions.
• Open sets and closed sets.
• Compactness and connectedness.
• Separation axioms (e.g., Hausdorff property).
• Homeomorphisms also preserve topological invariants such as Euler characteristic, dimensionality, and homotopy type.