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?

A topological invariant is usually defined as a property preserved under homeomorphisms.
There are topological properties, such as compactness and connectedness, that are preserved even under continuous functions.
Not all topological properties are preserved under continuous functions. For instance, the number of connected components (the simplest homology) is a topological invariant that is not preserved in general under continuous functions.
