T ₁ room

In topology and related areas of mathematics , T ₁ spaces are special topological spaces that have certain pleasant properties. The T ₁ axiom is an example of a separation axiom .

definition

Let X be a topological space . X is called T ₁ -space if for any two points each has a neighborhood in which the other does not lie. To delimit: In a T Raum-room only one of the two points must have such an environment, in a T₂-room the two environments must be disjoint. It is also said that a T ₁ space has a Fréchet topology . In this context, the term Fréchet space , which is a term from functional analysis , should be avoided .

properties

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

• X is a T ₁ space.
• X is a Kolmogoroff space and an R ₀ space .
• All single point sets in X are closed .
• Every finite set is closed.
• Any set with finite complement is open.
• Every elementary filter for any x only converges to x .
• For every subset S of X it holds that an element x from X is an accumulation point of S if and only if every open neighborhood of x contains an infinite number of elements.

The following implications always apply in topological spaces

separated ⇒ topologically distinguishable ⇒ disjoint

If the first arrow can be reversed, it is an R ₀ space , and precisely in a T ₀ space this also applies to the second implication. This shows that a topological space satisfies T ₁ if and only if it is both an R ₀ -space and a T ₀ -space.

Examples

The Zariski topology on an algebraic variety (in the classical sense) is T ₁ . To see this we consider a point with a local coordinate . The associated one-point set is the set of zeros of the polynomials . The point is thus closed. ${\displaystyle (c_{1},\ldots c_{n})}$ ${\displaystyle X-c_{1},\ldots ,X-c_{n}}$

For another example, consider the cofinite topology on a countable set, such as the set of integers . As an open set, we define exactly the empty set and the sets with finite complement. So you have all the shape with a finite set A . Now let x and y be two different points. The set is an open set that contains x and y does not. On the other hand, the element y does not contain x . It is therefore actually a T ₁ space. But this can also be deduced from the fact that one-element sets are closed. However, this space is not a T ₂ space. Because for two finite sets A and B holds what can never be empty. Furthermore, the set of even numbers is compact but not closed, which can never be the case in a T ₂ space. ${\displaystyle \mathbb {Z} }$ ${\displaystyle O_{A}=\mathbb {Z} \setminus A}$ ${\displaystyle O_{\{y\}}}$ ${\displaystyle O_{\{x\}}}$ ${\displaystyle O_{A}\cap O_{B}=O_{A\cup B}}$

More generally, for every topological space that satisfies the T ₁ axiom, its topology already includes the cofinite topology. The co-finite topology is thus the coarsest T ₁ topology on a set.

literature

• Boto von Querenburg : Set theoretical topology (= Springer textbook ). 3rd, revised and expanded edition. Springer, Berlin et al. 2001, ISBN 3-540-67790-9 .