## T0-spaces

Abbreviation: Top$_0$

### Definition

A $T_0$-space is a topological spaces $\mathbf{X}=\langle X,\Omega(\mathbf{X})\rangle$ such that

for every pair of distinct points in the space, there is an open set containing one but not the other: $x,y\in X\Longrightarrow\exists U\in\Omega(\mathbf{X})[(x\in U\mbox{ and }y\notin U)\mbox{ or }(y\in U\mbox{ and }x\notin U)]$

##### Morphisms

Let $\mathbf{X}$ and $\mathbf{Y}$ be $T_0$-spaces. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $f:X\rightarrow Y$ that is continuous:

$V\in\Omega(\mathbf{Y})\Longrightarrow f^{-1}[V]\in\Omega(\mathbf{X})$

Example 1:

### Properties

Classtype second-order yes no no

Remark: The properties given above use an $(\mathcal{E},\mathcal{M})$ factorization system with $\mathcal{E}=$ surjective morphisms and $\mathcal{M}=$ embeddings.

### References

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