"Some of these restrictions are given by the separation axioms."
Separation axioms are properties of topological spaces that describe how to separate points and subsets from each other. Separation axioms include T0, T1, T2, T3, and T4.
Topology basics: An introduction to the basic concepts of point-set topology, including open and closed sets, neighborhood, and continuity.
Separation properties: The different types of separation properties, including Hausdorff, regular, completely regular, normal, and Urysohn.
Hausdorff space: A topological space where any two distinct points can be separated by neighborhoods.
Regular space: A topological space where any two disjoint closed sets can be separated by neighborhoods.
Completely regular space: A topological space where any closed set is the intersection of neighborhoods of the set.
Normal space: A topological space where any two disjoint closed sets can be separated by open sets.
Urysohn space: A topological space where any two distinct points can be separated by a continuous function.
T0 and T1 spaces: Topological spaces where points can be distinguished by open sets (T1) or by the presence of a neighborhood (T0).
Compact spaces: Topological spaces where every open cover has a finite subcover.
Regular and normal compact spaces: Compact spaces that are also regular or normal.
Product spaces: The topology on the Cartesian product of two or more topological spaces.
Quotient spaces: Topological spaces obtained by identifying certain points of another space.
Metric spaces: Topological spaces with a distance function that satisfies certain axioms.
Separable spaces: Topological spaces with a countable dense subset.
Urysohn's Lemma: A fundamental result in topology that characterizes completely regular spaces.
Tietze Extension Theorem: A result that allows continuous functions defined on a closed subset of a normal space to be extended to the entire space.
Stone-Čech compactification: A construction that extends a given topological space to a compact space that has certain nice properties.
Alexandroff Extension: A construction that extends a given topological space to a Hausdorff space that has certain nice properties.
$T_0$: For any two distinct points, there is an open set containing one point but not the other.
$T_1$: For any two distinct points, there are disjoint open sets containing each point.
$T_2$ or Hausdorff: For any two distinct points, there are disjoint open sets containing each point.
Regular: For any point and closed set not containing the point, there are disjoint open sets containing the point and the closed set.
Completely regular: For any point and closed set not containing the point, there is a continuous function that separates the point from the closed set.
Normal: For any two disjoint closed sets, there are disjoint open sets containing each set.
Completely normal: For any two disjoint closed sets, there is a continuous function that separates the sets.
Urysohn: A space is Urysohn if and only if it is completely regular and normal.
Tychonoff: For any point and closed set not containing the point, there is a continuous function that separates the point from the closed set.
Perfectly normal: A perfectly normal space is a normal space where any two disjoint closed sets can be separated by a zero set (i.e., a set that is the intersection of a countable family of open sets).
Countably normal: A countably normal space is a normal space where any countable collection of pairwise disjoint closed sets can be separated by open sets.
Collectionwise normal: A collectionwise normal space is a normal space where any countable collection of closed sets can be separated by open sets.
Completely Hausdorff: A completely Hausdorff space is a space where any two points can be separated by a closed set.
Almost Hausdorff: An almost Hausdorff space is a space where any two distinct points have disjoint closures.
Weakly Hausdorff: A weakly Hausdorff space is a space where any two distinct points have neighborhoods whose intersection is not the empty set.
Hyperconnected: A hyperconnected space is a space where the only closed sets are the empty set and the whole space.
Punctiform: A punctiform space is a space where any two distinct points have neighborhoods whose intersection is finite.
Ultratopological: An ultratopological space is a space in which every closed set is the intersection of a family of open sets.
"These are sometimes called Tychonoff separation axioms."
"The separation axioms are not fundamental axioms like those of set theory."
"These are... defining properties which may be specified to distinguish certain types of topological spaces."
"The separation axioms are denoted with the letter 'T'..."
"Increasing numerical subscripts denote stronger and stronger properties."
"The precise definitions of the separation axioms have varied over time."
"These [separation axioms] are sometimes called Tychonoff separation axioms, after Andrey Tychonoff."
"In topology and related fields of mathematics..."
"Some of these restrictions are given by the separation axioms."
"The separation axioms... may be specified to distinguish certain types of topological spaces."
"Especially in older literature, different authors might have different definitions of each condition."
"The separation axioms are denoted with the letter 'T' after the German Trennungsaxiom ('separation axiom')."
"Increasing numerical subscripts denote stronger and stronger properties."
"The separation axioms are denoted with the letter 'T'..."
"These [separation axioms] are... defining properties which may be specified to distinguish certain types of topological spaces."
"These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff."
"Some of these restrictions are given by the separation axioms."
"The separation axioms... in topology and related fields of mathematics..."
"Especially in older literature, different authors might have different definitions of each condition."