Sierpiński space

The Sierpiński - space is a topological space consisting of two points, in which a quantity is exactly open and not yet completed. It is the smallest space with non-discrete and non-trivial topology.

Definition

The Sierpiński the space underlying point set is; its open sets are and.

Relation to other topological spaces

Is an arbitrary amount, and a two-element size, then a subset corresponding to each function, and vice versa.

Too analogous role takes over in the case of continuous functions and open sets. Be an arbitrary topological space. For a continuous function is as defined for continuous functions that the inverse images of open sets are open. and. An interesting result. Indeed, this is an open subset of, and is uniquely determined by the constant.

The Sierpiński room is a cogenerator of the category of Kolmogorov spaces: Are continuous maps between two Kolmogorov - spaces and, then there exists a continuous map such that: Be this with, it is separated by at least an open neighborhood of, or vice versa ( as a Kolmogorov space ). Then gives the desired. In fact, the cogenerators the category of Kolmogorov - rooms are just all Kolmogorov - spaces containing a subspace that is homeomorphic to.