\chapter{Continuous functions}




\section{Continuous function (Topological space)}


%Students of real analysis typically get their first taste of rigorously defining continuous function by means of limits, and consequently, an epsilon-delta condition. This does the trick for real functions, 

%Closeness in the image implies closeness in the domain.
%We start by defining continuity at a point.


Recall that in our study of metric spaces we managed to obtain an equivalent definition of continuous functions in terms of open neighborhoods; we can port this as our definition of a countinuous function in a general topological setting.

\begin{definition}[Continuity at a point]
	Let $(X,\mathcal{T}_{X})$ and $(Y,\mathcal{T}_{Y})$ A function $f : X \to Y$ is \emph{continuous at $p \in X$} iff for any neighborhood $V_{f(p)} \subseteq Y$ of $f(p)$, there exists a neighborhood $U_p \subseteq X$ of $p$ such that $f(U_p) \subseteq V_{f(p)}$.
\end{definition}

As expected, for continuous functions of a metric space this condition holds for every point, so should it hold for all points in a topological space.

\begin{definition}[Continuous function]
Let $(X,\mathcal{T}_{X})$ and $(Y,\mathcal{T}_{Y})$ A \emph{continuous function} is a function $f : X \to Y$ that is continuous at all points of its domain. 
\end{definition}

Continuous functions therefore map every neighborhood of any image point $f(p)$ to a neighborhood of point $p$. Note that every open set (save the empty set) is a neighborhood of some point, so continuous functions simply map open sets of $(Y,\mathcal{T}_Y)$ from open sets of $(X,\mathcal{T}_X)$.

\begin{proposition}[Equivalent definition of a continuous function]
	Let $(X,\mathcal{T}_{X})$ and $(Y,\mathcal{T}_{Y})$ A function $f : X \to Y$ is \emph{continuous} iff for any $V$ open in $(Y,\mathcal{T}_Y)$, $f^{-1}(V)$ is open in $(X,\mathcal{T}_X)$.
\end{definition}

This perspective of a continuous function is the most general and perhaps one of the most powerful, since a topology is defined by its open sets.


Notably, we can recylce our fact that neighborhoods are always within open balls to reverse engineer the definition of a continuous function from real analysis to obrain the topological definition.


In real analysis, one proves that continuity of real functions is preserved under the 4 arithmetic operations and composition; the only operator that can be well defined for continuous functions on generic topological spaces is the composition operation and we will prove that continuity is closed under it.

\begin{proposition}
Let $f$ and $g$ be continuous functions, then $f \circ g$ is continuous.
\end{proposition}


- Equivalent definitions of continuous function
- clousre based,
- point based
- point based



\section{Homeomorphisms}

Like how group homomorphishms preserve a groups structure, continuous functions preserve neighborhood structure of a topology. This leads to the question; like how group isomorphisms show that two groups are 'algebraically equivalent', is there some class of function to show that two topological spaces are 'topologically equivalent' (in the sense that they have the 'isomorphic neighborhoods')?


\begin{definition}[Homeomorphism]
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}

Since topological spaces are defined by their open sets, homeomrophic topological spaces have indistinguishable topological properties.

\[ f( \partial_{X} U) = \partial_{Y} f(U) \]
\[ \mathrm{cl}_{Y}(f(U)) = f(\mathrm{cl}_{X}(U)) \]


This leads to the common mathematical joke that topologists can't tell the difference between a coffee mug and a donut; since a torus (donut) can be continuously deformed to a coffee mug, they are homeomorphic and hence have the same properties insofar as topology is concerned. 

Since homeomorphisms translate open sets of 2 topological spaces between eachother, any property that can be expressed purely by open sets is preserved under a homeomorphism. Here are some examples of types of properties conserved under homeomorphisms.

\begin{itemize}
	\item Connectedness properties
	\item Compactness properties
	\item Separation properties
	\item Countablility properties
	\item Metrizability properties
\end{itemize}




\section{Embeddings}




Sometimes mathematicians realize that the topological space they're working with is just a subspace, or at least homeomorphic to one.

Think of Pac-man's world, where the top of the screen warps to the bottom and the left warps to the right. This is homeomorphic to a torus, so Pac-man is living within a torus-shaped 'planet' in $\mathbb{R}^3$; it is a subspace that is 'embedded' within $\mathbb{R}^3$.

\begin{definition}[Embedding]
a continuous function between topological spaces is an embedding iff it is a homeomorphism from X to a topological subspace of Y
\end{definition}

-topological properties