Topology and Continuity concepts

Let $S$ be a subset of a metric space $M$

• $S$ is closed if it contains all its limits.
• $S$ is open if for each $p\in S$ there exists an $r>0$ such that the open ball $B(p,r)$ is entirely contained in $S$
• The complement of an open set is closed and vice versa.

The topology of $M$ is the collection $\mathcal{T}$ of all open subsets of $M$.

$\mathcal{T}$ has the following properties

• It is closed under arbitrary union of open sets
• It is closed under finite intersections
• $\emptyset, M$ are open sets.

Corollary

• arbitrary intersection of closed sets is closed
• finite union of closed sets is closed
• $\emptyset, M$ are closed sets.

A metric space $M$ is complete if each Cauchy sequence in $M$ converges to a limit in $M$.  

• $\mathbb{R}^n$ is complete

Every compact set is closed and bounded

Continuity of function $f: M \rightarrow N$

• The pre-image of each open set in $N$ is open in $M$ 
• Preserves convergence sequences under the transformation, i.e.
  $$f( \lim x_n) = \lim f(x_n)$$ for every convergent sequence $\{x_n\}$