◀ ▲ ▶Branches / Topology / Definition: Topological, Continuous, Open, and Closed Invariants
Definition: Topological, Continuous, Open, and Closed Invariants
A property of a topological space $(X,\mathcal O_X)$ is called
- a topological invariant, if any other space $(Y,\mathcal O_Y)$ possesses the same property, whenever there is a homeomorphism $f:X\to Y,$
- a continuous invariant, if the image $f[X]$ of a continuous function $f:X\to Y$ possesses the same property in $(Y,\mathcal O_Y),$
- an open invariant, if the image $f[X]$ of an open function $f:X\to Y$ possesses the same property in $(Y,\mathcal O_Y),$
- a closed invariant, if the image $f[X]$ of a closed function $f:X\to Y$ possesses the same property in $(Y,\mathcal O_Y).$
Instead of using the term "invariant", we may also use the terms topological property, continuous property, open property, and closed property.
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References
Bibliography
- Steen, L.A.;Seebach J.A.Jr.: "Counterexamples in Topology", Dover Publications, Inc, 1970
- Jänich, Klaus: "Topologie", Springer, 2001, 7th Edition
