T1 space

In topology and related areas of mathematics T1 - spaces are special topological spaces that hold some nice features. The T1 - axiom is an example of a separation axiom.

Definition

Let X be a topological space. X is called a T1 - space if each has a neighborhood for any two points in which the other is not. To distinguish: For a T ₀ - room only one of the two points must have such an environment, when a T ₂ - room the two environments must be able to be chosen disjoint. It is also said that a T1 space has a Fréchet topology. To be avoided in this context the term Fréchet space which is a concept from functional analysis.

Properties

Let X be a topological space. The following are equivalent:

  • X is a T1 - space.
  • X is a Kolmogorov - room and an R0 - space.
  • All one-point sets in X are complete.
  • Every finite set is closed.
  • Plenty of finite complement is open.
  • Each elementary filters at any x converges to x.
  • For each subset S of X, that an element x of X if and only a limit point of S if every open neighborhood of x contains infinitely many elements.

In topological spaces always have the following implications hold

If the first arrow can be reversed, it is an R0 - space, exactly this is true in a T0- space for the second implication. Thus we see that a topological space if and only T1 satisfied if he is both an R0 - space and a T0- space.

Examples

The Zariski topology on an algebraic variety ( in the classical sense ) is T1. To see this, consider a point with local coordinates. The associated one-point set is the set of zeros of polynomials. The point is thus completed.

For another example, consider the kofinite topology on a countable set, such as the set of integers. As an open set, we define precisely the empty set and the quantities with finite complement. So you all have the shape of a finite set A. Now let x and y be two distinct points. The amount is an open set that contains x and y not. On the other hand, the element y contains x but not. Thus, it actually is a T1 - space. This can be concluded as well as from the fact that one element amounts have been completed. This space, however, is not a T2 - space. Because for two finite sets A and B is true, what can never be empty. Further, the set of even numbers is compact, but not complete, which can never be the case in a T2 zone.

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