A continuous map is a function between two topological spaces that preserves the notion of closeness, meaning the preimage of every open set is open. This concept is fundamental in topology as it allows for the comparison of different spaces by examining how they can be transformed into one another without 'tearing' or 'gluing' points together.

Continuous Maps

A continuous map is a function between topological spaces that preserves the notion of closeness, meaning the preimage of every open set is open. It is a fundamental concept in topology, ensuring that the structure of the spaces is maintained under the mapping, which is crucial for understanding homeomorphisms and other topological properties.

continuous map

Topological spaces are a fundamental concept in mathematics, providing a framework for discussing continuity, convergence, and boundary in a more general sense than metric spaces. They consist of a set of points along with a collection of open sets that satisfy specific axioms, allowing for the exploration of properties like compactness and connectedness without the need for a defined distance function.

topological spaces

notion of closeness

preimage of every open set

An open set is a fundamental concept in topology, characterized by the property that for any point within the set, there exists a neighborhood entirely contained within the set. This concept is crucial for defining and understanding continuity, limits, and convergence in a topological space.

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