Separation axioms are properties that a topological space can satisfy depending on the degree to which different points or closed sets can be separated by means of the open sets of a certain topology.
Summary
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- 1 Increasing levels
- 2 Topology
- 3 Some axioms of separation
- 1 T 0or Kolmogórov spaces
- 2 T 1spaces or Fréchet space
- 3 T 2or Hausdorff spaces
- 4 T 3or regular spaces
- 5 Completely regular spaces and T 3.5or Tikhonov spaces
- 6 T 4or normal spaces
- 4 Sources
Increasing levels
There are several increasing levels of separation that can be requested from a topological space. They are usually named with the letter T (for Trennung , German separation) and a convenient subscript. Thus a hierarchy of spaces appears, among which we must highlight the T 2 spaces or the Hausdorff space , the T 3 or regular spaces and the T 4 or normal spaces .
Except for T 0 , T 1 and T 2 , the names of the separation axioms are not properly standardized.
Topology
The topology, in its generality, admits little useful topological structures: let’s think of a set X with more than one element, endowed with the trivial topology ( ie its only open ones are Ø and all X ). This topology does not contain openings that allow us to distinguish two different points topologically: both points share the only possible environment . Looking at the open environments of each point, it is impossible to distinguish them. We say that, for topological purposes, X is no different from a single-point set endowed with trivial topology.
The axioms separation are requirements on topology space to ensure the existence of a sufficient number of open sets to distinguish topologically different points. The different degrees to which it takes shape, this requirement is reflected in the various axioms of separation.
Some axioms of separation
Kolmogórov T 0 spaces
A topological space X is called [[space T 0 if and only if for any pair of points s, t of X there is an open that contains one of the points and does not contain the other point.
An equivalence to this property is as follows: if s and t are elements of space X such that the closure of {x \} and the closure of {y \} are equal then x = y
T 1 spaces or Fréchet space
A topological space X is said T 1 if and only if for any pair of points s, t of X there are a pair of open sets A and B, such that s is in A, but not in B, and furthermore t is in B , but not in A.
An important equivalence is that X is T 1 if and only if the subsets of X formed by a single point are closed.
T 2 or Hausdorff spaces
A topological space X is Hausdorff or T 2 if and only if for any pair of points x, and in X there exists a pair of disjoint openings that contains one a and the other a y.
These spaces are especially important because in addition to supplying a large number of examples (all metric spaces T 2 are, they have strong properties such as the convergence of a sequence or a filter , if it exists, is unique.
T 3 or regular spaces
A topological space X is regular if it is T 1 and for each point x of X and any closed F subset of X such that x does not belong to F. Then there are environments
U x and U F such that their intersection is empty. That is, we can separate closed points.
Completely regular spaces and T 3.5 or Tikhonov spaces
A topological space X is completely regular if for each point x of X and any closed F subset of X such that x does not belong to F there exists a continuous function F: → <math [0, 1] such that f (x) = 0 and f (F) =.
A topological space X is from Tikonov if it is T 1 <math> and completely regular. It can also be designated as a completely regular Hausdorff space.
T 4 or normal spaces
A topological space X is normal if it is T 1 and for each pair of closed F and G subsets of X with empty intersection there are environments that contain them U and W and such that their intersection is equal to the empty set.
In other words, we can separate all the closed spaces. In particular metric spaces are normal.
