# Topological invariants

Do continuous maps necessarily preserve topological invariants? Or is it necessary for the maps to be homeomorphisms? Are there simple examples where continuous maps do not preserve these invariants?

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Note that there is a continuous map from any topological space to a single point, so no invariant that can distinguish non-points from points will be preserved in general. A homeomorphism will preserve every invariant (by the definition of invariant, as pointed out by lhf).

However, many topological invariants (such as the fundamental group and homology) are preserved by homotopy equivalences, which are not homeomorphisms in general, so there is a middle ground.

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