# General topology
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## Topological spaces
- topology
\begin{definition}
A \emph{topology on a set $X$} is a set $\mathcal{T}$ of subsets of $X$ such that:
- $X$ and $\emptyset$ are in $\mathcal{T}$
- $\mathcal{T}$ is closed under finite intersections
- $\mathcal{T}$ is closed under countable unions

\[ \mathcal{T} \subseteq \mathcal{P}(X) \text{ is a topology on }X \iff X,\emptyset \in \mathcal{T} \land [  \bigcap^{n}_{i=0}U_i \in \mathcal{T} ] \land [ \bigcup^{\infty}_{i=0} U_i \in \mathcal{T} ] \]
\end{definition}
- discrete topology
- indiscrete topology
- topological space
A \emph{topological space} is an ordered pair $(\mathcal{T},X)$ of a set $X$ and a topology $\mathcal{T}$ on $X$ denoted as . Elements of $X$ are referred to as \emph{points}.

- interior
- open set
- closed set
- clopen set
- cofinite set
- cofinite topology
### Examples of topological spaces
- T\_0 space (Kolmogorov space)
\begin{definition}
A \emph{$T_0$ space} is a topological space such that for every distinct pair of points, at least 1 point in the pair has a neighborhood not containing the other point.
\end{definition}
- T\_1 space (Topological Fréchet space)
\begin{definition}
A \emph{$T_1$ space} is a topological space such that for every distinct pair of points, both points in the pair has a neighborhood not containing the other point.
\end{definition}
- T\_2 space (Hausdorff space)
\begin{definition}
A \emph{$T_2$ space} is a topological space such that for every distinct pair of points, there exists a pair of neighborhoods of both points which is mutually exclusive.
\end{definition}
- finite frechet spaces are discrete topological spaces
- simply conencted space
### Euclidean topology
- Euclidean topology
### Basis
- basis (topological space)
- when does a basis generate a topology? Existence of topology for a basis
- generating a basis for given topology
- identical topologies by comparing bases
### Topological subspaces
- topological subspace

## Limit points
- boundary
- limit point
- set of a set's limit points is closed
- closure
- neighborhood
- dense set
- open cover
- subcover

## Continuous maps
- continuous function (topology)

## Metric spaces
- distance function
- metric space
### Examples of metric spaces
- euclidean metric
- chebyshev metric
- taxicab metric


## Homeomorphisms
- homeomorphism
\begin{definition}
A \emph{homeomorphism} between two topological spaces $T$ and $U$ is a bijective function $f : T \to U$ such that both $f$ and $f^{-1}$ are continuous.
\end{definition}






- Jordan curve theorem (JCT)
- Urysohn's lemma