We know that if a property is defined in terms of open sets then the property is preserved under a homeomorphism. Is there a name for such a property?
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1a topological property/invariant. – Ittay Weiss Jul 30 '14 at 08:22
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I think it is called topological invariant. By definition, a topological invariant is is a property of a topological space which is invariant (i.e., preserved) under homeomorphisms, see here.
Dietrich Burde
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1I think that is not quite what the OP is asking, although I could be wrong. For example, "cardinality of the underlying set" is a topological invariant which is not defined in terms of open sets. Properties of topological spaces defined only in terms of their open sets are properties of their underlying locales (http://en.wikipedia.org/wiki/Pointless_topology), so really should be called localic properties, I guess. – Qiaochu Yuan Jul 30 '14 at 08:47