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How does a homeomorphism use continuous functions to connect topologically equivalent shapes?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

A homeomorphism connects two topological spaces by establishing a one-to-one mapping that preserves the underlying structure of the objects. For two shapes to be considered topologically equivalent, there must exist a function $h : X \to Y$ that satisfies three specific criteria:12

One-to-one correspondence: Every point in space $X$ maps to a unique point in space $Y$ , and vice versa.12
Continuity: The function $h$ is continuous, meaning it preserves the "neighborhoods" of points; nearby points in $X$ are mapped to nearby points in $Y$ .23
Continuous inverse: There exists an inverse function $h^{- 1}$ that is also continuous, ensuring that the mapping can be "undone" without tearing or breaking the structure.12

Because both the function and its inverse are continuous, the homeomorphism ensures that topological properties—such as connectedness or compactness—remain unchanged, effectively treating the shapes as identical from a topological perspective.13