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 \(\{x_n\}\)