T1 space

From Wikipedia, the free encyclopedia

Topological spaces in separation axiom
Kolmogorov (T0) version
T0 | T1 | T2 | T2½ | completely T2 T3 | T3½ | T4 | T5 | T6

In topology and related branches of mathematics, T₁ spaces and R₀ spaces are particular kinds of topological spaces. The T₁ and R₀ properties are examples of separation axioms.

Contents 1 Definitions 2 Properties 3 Examples 4 Generalisations to other kinds of spaces

[edit] Definitions

Let X be a topological space and let x and y be points in X. We say that x and y can be separated if each lies in an open set which does not contain the other point.

• X is a T₁ space if any two distinct points in X can be separated.

• X is a R₀ space if any two topologically distinguishable points in X can be separated.

A T₁ space is also called an accessible space or a Fréchet space and a R₀ space is also called a symmetric space. (The term Fréchet space also refers to an entirely different notion from functional analysis. For this reason, the term T₁ space is preferred. There is also a notion of a Fréchet-Urysohn space as a type of sequential space.)

[edit] Properties

Let X be a topological space. Then the following conditions are equivalent:

• X is a T₁ space.
• X is a T₀ space and an R₀ space.
• Points are closed in X; i.e. given any x in X, the singleton set {x} is a closed set.
• Every finite set is closed.
• Every cofinite set of X is open.
• The fixed ultrafilter at x converges only to x.
• For every point x in X and every subset S of X, x is a limit point of S if and only if every open neighbourhood of x contains infinitely many points of S.

Proof. Suppose singletons are closed in X. Let S be a subset of X and x a limit point of S. Suppose there is an open neighbourhood U of x that contains only finitely many points of S. Then U \ (S \ {x}) is an open neighbourhood of x that does not contain any points of S other than x. (Here is where we use the fact that singletons are closed.) This contradicts the fact that x is a limit point of S. Thus, every open neighbourhood of x contains infinitely many points of S. Conversely, suppose there is a point x in X such that the singleton {x} is not closed. Then there is a point y ≠ x in the closure of {x}. We claim that any open neighbourhood U of y contains x. For suppose not; then the complement of U in X would be a closed set containing x, and the closure of {x} would be contained in the complement of U. Since y is in the closure of {x}, this would force y not to be in U, contradicting the fact that U is a neighbourhood of y. We have shown that y is a limit point of S = {x}. But it is clear that X is a neighbourhood of y that does not contain infinitely many points of S.

Let X be a topological space. Then the following conditions are equivalent:

• X is an R₀ space.
• Given any x in X, the closure of {x} owns only the points that x is topologically indistinguishable from.
• The fixed ultrafilter at x converges only to the points that x is topologically indistinguishable from.
• The Kolmogorov quotient of X (which identifies topologically indistinguishable points) is T₁.
• Every open set is the union of closed sets.

In any topological space we have, as properties of any two points, the following implications

separated ⇒ topologically distinguishable ⇒ distinct

If the first arrow can be reversed the space is R₀. If the second arrow can be reversed the space is T₀. If the composite arrow can be reversed the space is T₁. Clearly, a space is T₁ if and only if it's both R₀ and T₀.

Note that a finite T₁ space is necessarily discrete (since every set is closed).

[edit] Examples

• Sierpinski space is a simple example of a topology that is T₀ but is not T₁.
• The overlapping interval topology is a simple example of a topology that is T₀ but is not T₁.

• The cofinite topology on an infinite set is a simple example of a topology that is T₁ but is not Hausdorff (T₂). This follows since no two open sets of the cofinite topology are disjoint. Specifically, let X be the set of integers, and define the open sets O_A to be those subsets of X which contain all but a finite subset A of X. Then given distinct integers x and y:

• the open set O_{x} contains y but not x, and the open set O_{y} contains x and not y;
• equivalently, every singleton set {x} is the complement of the open set O_{x}, so it is a closed set;

so the resulting space is T₁ by each of the definitions above. This space is not T₂, because the intersection of any two open sets O_A and O_B is O_A∪B, which is never empty. Alternatively, the set of even integers is compact but not closed, which would be impossible in a Hausdorff space.

• The above example can be modified slightly to create the double-pointed cofinite topology, which is an example of an R₀ space that is neither T₁ nor R₁. Let X be the set of integers again, and using the definition of O_A from the previous example, define a subbase of open sets G_x for any integer x to be G_x = O_{{x, x+1}} if x is an even number, and G_x = O_{{x-1, x}} if x is odd. Then the basis of the topology are given by finite intersections of the subbasis sets: given a finite set A, the open sets of X are

The resulting space is not T₀ (and hence not T₁), because the points x and x + 1 (for x even) are topologically indistinguishable; but otherwise it is essentially equivalent to the previous example.

• The Zariski topology on an algebraic variety is T₁. To see this, note that a point with local coordinates (c₁,...,c_n) is the zero set of the polynomials x₁-c₁, ..., x_n-c_n. Thus, the point is closed. However, this example is well known as a space that is not Hausdorff (T₂). The Zariski topology is essentially an example of a cofinite topology.

[edit] Generalisations to other kinds of spaces

The terms "T₁", "R₀", and their synonyms can also be applied to such variations of topological spaces as uniform spaces, Cauchy spaces, and convergence spaces. The characteristic that unites the concept in all of these examples is that limits of fixed ultrafilters (or constant nets) are unique (for T₁ spaces) or unique up to topological indistinguishability (for R₀ spaces).

As it turns out, uniform spaces, and more generally Cauchy spaces, are always R₀, so the T₁ condition in these cases reduces to the T₀ condition. But R₀ alone can be an interesting condition on other sorts of convergence spaces, such as pretopological spaces.