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.

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.

There are two notions of compactness, they turns out to be equivalent for metric spaces.

Let $X$ be a metric space, $K \In X$ a subset.

• sequential compactness: we say $K$ is compact, if every sequence in $K$ has a convergent subseq.
• compactness: any open cover of $K$ admits a finite subcover.

The two notions turns out are equivalent, see https://courses.wikinana.org/math104-f21/compactness We will follow Pugh to give a proof. See also Rudin Thm 2.41