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Which conditions must a function satisfy to be a homeomorphism?

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topology Because both a doughnut and a coffee cup have one hole (handle), they can be mathematically, or topologically, transformed into one another without cutting them in any way. For this reason, it has often been joked that topologists cannot tell the difference between a coffee cup and a doughnut.

In topology, a function $h : X \to Y$ is defined as a homeomorphism if it establishes a topological equivalence between two objects. To qualify as a homeomorphism, the function must satisfy the following conditions:

One-to-one correspondence: The function must be a bijection, meaning it is both one-to-one and onto, ensuring every element in $X$ corresponds to exactly one element in $Y$ and vice versa.12
Continuity: The function $h$ must be continuous, effectively preserving the structure of "neighbourhoods" by mapping nearby points in $X$ to nearby points in $Y$ .2
Continuous inverse: There must exist an inverse function $h^{- 1} : Y \to X$ that is also continuous. This ensures that the mapping can be "undone" while maintaining the topological structure in both directions.12

When these conditions are met, the spaces $X$ and $Y$ are considered topologically equivalent, meaning they share the same topological properties, such as connectedness and compactness.12