Topological indistinguishability: Difference between revisions
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==Definition== | ==Definition== | ||
'''Topological | '''Topological indistinguishability''' is an equivalence relation on any [[topological space]]. For a topological space <math>X</math>, two (possibly equal, possibly distinct) points <math>x,y \in X</math> are termed '''topologically indinstinguishable''' if the following equivalent conditions hold: | ||
# The [[closure]]s of the singleton sets <math>\{ x \}</math> and <math>\{ y \}</math> are equal. | # The [[closure]]s of the singleton sets <math>\{ x \}</math> and <math>\{ y \}</math> are equal. | ||
Latest revision as of 02:14, 28 January 2012
Definition
Topological indistinguishability is an equivalence relation on any topological space. For a topological space , two (possibly equal, possibly distinct) points are termed topologically indinstinguishable if the following equivalent conditions hold:
- The closures of the singleton sets and are equal.
- Every closed subset containing contains and every closed subset containing contains .
- Every open subset containing contains and every open subset containing contains .
Two distinct points that are not topologically indistinguishable are termed topologically distinguishable.
Related notions
- The Kolmogorov quotient of a topological space is the quotient by the equivalence relation of topological indistinguishability, and it is given the T0 topology.