If two curves have identical start and end points, one may ask if curve A can be continuously deformed into curve B and vice versa (like moving a piece of string). Such a deformation is known as a \emph{homotopy}.

\begin{definition}[Homotopy]
Two functions are \emph{homotopic} iff there exists some continuous function $H : X \times [0,1] \to Y$ such that $H(x,0)=f(x) , H(x,1)=g(x)$. Such a $H$ is called a \emph{homotopy}.
\end{definition}

\begin{definition}[Topological group]
A \emph{topological group} is a triple $(G,\mathcal{T},\cdot)$ where
\begin{itemize}
	\item $(G,\cdot)$ is a group
	\item $(G,\mathcal{T})$ is a topological space
	\item $\cdot$ and $^{-1}$ are continuous functions, where $G\times G$ has the product topology
\end{itemize}
\end{definition}
They're basically groups that are also topological spaces.
- homotopy
- fundamental group

we can 'glue' or 'concatenate' loops together

\begin{definition}[Loop concatenation operation at basepoint $p$]
\end{definition}

Let's consider equivalence classs of loops (with the same basepoint) that are homotopic to eachother in the topological space. One can see this as being quite handy with even a basic intuition; we can see which loops can be continuously deformed between themselves and if they can't, there must be some topological phenomenon intervening.


If we consider loop concatenation as an operation, this actually defines a group! Like always, we need to ensure that our group operation is well defined, so we check that if $[\gamma_1][\gamma_2]=[ \gamma_3]$, then for any $\eta \in [\gamma_1], \eta_2 \in [\gamma_2]$ then $\gamma_2 \cdot \gamma_1 \in \gamma_3$. Furthermore we want this operation to satisfy the laws of a group.

This all works out, and is called the \emph{fundamental group} of that topological space.

\begin{definition}[Fundamental group of a topological space with basepoint $p$]
	$\pi_1 (X,p)$
\end{definition}



- homology
- Betti numbers


\chapter{Simply connected spaces}

Connected spaces are important in real analysis for results such as the intermedate value theorem and  the first derivative test, however complex analysis also relies heavily on another notion of connectedness; simply connected spaces. They are apart of the assumptions for Cauchy's theorem to apply; the most important theorem in the field.

Simply put, they are spaces where there are no 'holes'.

\begin{definition}[Simply connected space]
			<p>Topological space with no holes, characterized by the ability to continuously transform any loop around a point</p>

A \emph{simply connected space} is a topological space that is path connected
			<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>

\end{definition}
	




- Jordan curve theorem $\mathbb{R}^2$
- Generalized Jordan curve theorem
-