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Homeomorphic spaces have the same cardinality of points (i.e., they have the same 'number of points').

Homeomorphism preserves the property of connectedness; if one space is connected, so must the other be.

A compact space is homeomorphic only to other compact spaces.

The property of being orientable is maintained under homeomorphism.

Homeomorphic spaces must have the same topological dimension.

Homeomorphism maps boundary points of one space to boundary points of the other space. Thus, the boundary structure is preserved.

The genus, or the number of 'holes' in a space, is an invariant under homeomorphism.

Homeomorphisms induce isomorphisms on homotopy groups, thereby preserving the homotopy type.

Homeomorphism only occurs between spaces that are either both Hausdorff or both non-Hausdorff.

Local topological properties like being locally Euclidean or having a local product structure are preserved by homeomorphisms.

The embedding of a topological space into another is preserved under homeomorphism.

Path-connectedness is an invariant of homeomorphisms; homeomorphic spaces are either both path-connected or not.

The continuity of functions between spaces is preserved by homeomorphisms.

Betti numbers, which indicate the maximum number of cuts that a space can have without becoming disconnected, are homeomorphism invariants.