## Topological spaces

Abbreviation: Top

### Definition

A topological space is a structure $\mathbf{X}=\langle X,\tau\rangle$, where $\tau=\Omega(\mathbf{X})\subseteq \mathcal P(X)$ is a collection of subsets of $X$ called the open sets of $\mathbf{X}$ such that

any union of open sets is open: $\mathcal{U}\subseteq\Omega(\mathbf{X})\Longrightarrow\bigcup\mathcal{U}\in\Omega(\mathbf{X})$

any finite intersection of open sets is open: $U,V\in\Omega(\mathbf{X})\Longrightarrow U\cap V\in\Omega(\mathbf{X})$ and $X\in\Omega(\mathbf{X})$

Remark: Note that the union of an empty collection is empty, so $\emptyset\in\Omega(\mathbf{X})$.

The set of closed sets of $\mathbf{X}$ is $K(\mathbf{X})=\{X-U\mid U\in\Omega(\mathbf{X})\}$.

##### Morphisms

Let $\mathbf{X}$ and $\mathbf{Y}$ be topological 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 yes yes

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