====== Lecture 13: Continuous function.  ======

Last time we had some basic notion of metric space (a set with the notion of distances), and topological space (which is a set with some notion of which subsets are open). 

Today, we will define continuous functions (or rather continuous maps) between metric spaces and between topological spaces. 

** Def 1 **: Let $(X, d_X)$ and $(Y, d_Y)$ be two metric spaces, a map $f: X \to Y$ is continuous, if for every convergent sequence $p_n \to p$ in $X$, we have convergent sequence $f(p_n) \to f(p)$.

** Def 2 **: ($\epsilon-\delta$: Let $(X, d_X)$ and $(Y, d_Y)$ be two metric spaces, a map $f: X \to Y$ is continuous, if for every $x \in X$, and every $\epsilon > 0$, we have $\delta>0$, such that $f( B_\delta(x)) \In B_\epsilon(f(x))$. 

** Def 3 **: Let $X, Y$ be two topological spaces, we say a map $f: X \to Y$ is continuous, if for every open sets $V \In Y$, we have $f^{-1}(V)$ is open in $X$. 

We can discuss later, why the three definitions are equivalent. 

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Homeomorphism: if $f: X \to Y$ is continuous and a bijection, and if $f^{-1}: Y \to X$ is also continuous, then we say $f$ is a homeomorphism.  

Topologically, we cannot distinguish spaces that are homeomorphic.