\part{Connected spaces}


\chapter{Connected spaces}


The idea of 'connectedness' captures the property of a topology where it cannot be split

\begin{definition}
A \emph{connected space} $(X,\mathcal{T})$ is a topological space such that the only union of a pair of disjoint open sets is the space with the emptyset.
\end{definition}



The following theorem provides an alternative but equivalent definition of connected spaces that is often easier to work with.
\begin{theorem}
A \emph{connected space} $(X,\mathcal{T})$ is a topological space that.
\end{theorem}

\begin{definition}
A \emph{connected space} $(X,\mathcal{T})$ is a topological space .
\end{definition}

Euclidean topological space is connected.


Intermediate value theorem

Let $f : [0,1] \to [0,1]$ be a continuous function, then $f$ has a fixed point.

\chapter{Path connected space}
\begin{definition}
A \emph{path connected space} $(X,\mathcal{T})$ is a topological space such that for any pair of distinct points $x,y \in X$, there exists a continuous function $f : [0,1] \to X$ where $f(0)=x$ and $f(1)=y$. Essentially, any two points in a space has a continuous path from one another.
\end{definition}

- path connected spaces are connected spaces

- Topologist's sine curve; counterexample that connected spaces aren't always path connected

\chapter{Simply connected spaces}
- Allude to the importance in complex analysis
- simply connected space


			<h3>Simply connected space</h3>
			<p>Topological space with no holes, characterized by the ability to continuously transform any loop around a point</p>
			<p>\(X \text{ is simply connected } \iff\)</p>
			<p>\(X \text{ is path-connected}\)</p>
			<p>\(\forall \mathcal{C} : [t_0,t_1] \to X ( \mathcal{C} \text{ is a closed simple curve} \implies \exists f \ell (f(\mathcal{C}) = 0 )\)</p>

\chapter{Locally connected spaces}
- Locally connected




