A homeomorphism is a continuous function between two topological spaces that has a continuous inverse, essentially providing a way to show that two spaces are 'the same' in a topological sense. This concept plays a crucial role in understanding the properties of spaces, as it indicates that these spaces can be transformed into one another without tearing or gluing, preserving their topological characteristics. The significance of homeomorphisms can be seen in various mathematical contexts, from defining equivalence classes of shapes to analyzing structures like manifolds and submanifolds.
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