## T1-spaces

Abbreviation: Top$_1$

### Definition

A \emph{$T_1$-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 a pair of open sets containing each point but not the other: $x,y\in X\Longrightarrow\exists U,V\in\Omega(\mathbf{X})[x\in U\setminus V\mbox{ and }y\in V\setminus U]$

##### Morphisms

Let $\mathbf{X}$ and $\mathbf{Y}$ be $T_1$-spaces. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $f:X\rightarrow Y$ that is \emph{continuous}:

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

### Definition

A \emph{$T_1$-space} is a topological spaces $\mathbf{X}=\langle X,\Omega(\mathbf{X})\rangle$ such that all

singleton subsets are closed: $X\setminus\{x\}\in\Omega(\mathbf{X})$

Example 1:

### Properties

Classtype second-order yes yes yes

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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