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

While homeomorphisms preserve topological properties, they do not necessarily preserve metrical properties like distances and angles.

The property of being orientable is maintained under homeomorphism.

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

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

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

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

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

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

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

A compact space is homeomorphic only to other compact spaces.

The continuity of functions between spaces is preserved by homeomorphisms.

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.