Topological property

In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space "X" possesses that property every space homeomorphic to "X" possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
A common problem in topology is to decide whether two topological spaces are homeomorphic or not. To prove that two spaces are "not" homeomorphic, it is sufficient to find a topological property which is not shared by them.
Common topological properties.
Separation.
Note that some of these terms are defined differently in older mathematical literature; see history of the separation axioms.
Non-topological properties.
There are many examples of properties of metric spaces, etc, which are not topological properties. To show a property formula_16 is not topological, it is sufficient to find two homeomorphic topological spaces formula_17 such that formula_6 has formula_16, but formula_20 does not have formula_16.
For example, the metric space properties of boundedness and completeness are not topological properties. Let formula_22 and formula_23 be metric spaces with the standard metric. Then, formula_17 via the homeomorphism formula_25. However, formula_6 is complete but not bounded, while formula_20 is bounded but not complete.
References.
[2] Simon Moulieras, Maciej Lewenstein and Graciana Puentes, Entanglement engineering and topological protection by discrete-time quantum walks, Journal of Physics B: Atomic, Molecular and Optical Physics 46 (10), 104005 (2013).
https://iopscience.iop.org/article/10.1088/0953-4075/46/10/104005/pdf