nLab symmetric topological space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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(0,1)(0,1)-Category theory



A topological space (X,τ)(X,\tau) is called a symmetric topological space or an R 0R_0-topological space if for all elements xXx \in X and yXy \in X, if for all open sets UτU \in \tau, xUx \in U implies that yUy \in U, then for all open sets UτU \in \tau, yUy \in U implies that xUx \in U. Or equivalently, if its specialization preorder is an equivalence relation.


A T 1T_1-topological space is a symmetric topological space which is also a Kolmogorov topological space. The quotient of a symmetric topological space (X,τ)(X, \tau) by its specialization order equivalence relation is a T 1T_1-topological space, the Kolmogorov quotient of XX.


Every set XX with its power set 𝒫(X)\mathcal{P}(X) is a symmetric topological space with respect to the specialization order equivalence relation defined as

xyP𝒫(X).P(x)P(y)x \equiv y \coloneqq \forall P \in \mathcal{P}(X).P(x) \iff P(y)

for all xXx \in X and yXy \in X.

preorderpartial orderequivalence relationequality
topological spaceKolmogorov topological spacesymmetric topological spaceaccessible topological space

Last revised on July 16, 2023 at 20:22:14. See the history of this page for a list of all contributions to it.