Topologicallysuch

Topologicallysuch is a term occasionally encountered in informal discussions of topology to denote properties of spaces that depend only on the topological structure, not on any particular representation. The label is not standard in formal literature, but it can be used to illustrate how certain features are invariant under homeomorphisms.

Formally, a property P of a topological space X is called topologicallysuch if X has P whenever it is homeomorphic to Y that has P; equivalently, P is invariant under homeomorphisms. In this sense topologicallysuch properties are the classic topological properties, shedding dependence on metrics or coordinates.

Common examples include connectedness, compactness, separability, and various separation axioms such as Hausdorffness. Countability properties like first- and second-countability, as well as metrizability, also survive homeomorphisms. These properties can often be characterized by invariants of the lattice of open sets, neighborhoods, or by functorial constructions that are preserved by homeomorphisms.

Because topologicallysuch is not a formal term with a single fixed definition, usage varies by author. In teaching, it can help distinguish between features that are intrinsic to the topology and those that depend on extra structure such as a metric. Despite its informality, the idea captures a core principle: meaningful properties are preserved by homeomorphisms.

See also topology, homeomorphism, topological invariant, topological property.