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#### Abbreviation: Top_{0}

### Definition

A *T*_{0}-space is a topological space *X* = (*X*,*Ω*(*X*)) such that

for every pair of distinct points in the space, there is an open set containing one but not the other: *x*,*y* ∈ *X* ⇒ ∃*U* ∈ *Ω*(*X*)[(*x* ∈ *U* and *y* ∉ *U*) or (*y* ∈ *U* and *x* ∉ *U*)].

### Morphisms

Let *X* and *Y* be *T*_{0}-spaces.
A morphism from *X* to *Y* is a function *f* : *X*→*Y* that is *continuous*:
*V* ∈ *Ω*(*Y*) ⇒ *f*^{−1}[*V*] ∈ *Ω*(*X*).

### Some results

### Examples

### Properties

**Remark**:
The properties given above use an (*E*,*M*) factorization system with *E* = surjective morphisms and
*M* = embeddings.

### Subclasses

T1-spaces

### Superclasses

Topological spaces

### References

http://www.wikipedia.org/wiki/Topology_glossary