Advanced Calculus

Advanced Calculus as taught in Fitzpatrick's book.

Topology

Closed

The folllowing definitions of Closed Sets are equivalent.

• (Order)
• (Sequences) A set \(\displaystyle S\) is clsoed if it contains all its limit points. That is \(\displaystyle \forall \{x_i\} \subseteq S\), \(\displaystyle \{x_i\} \rightarrow x_0 \implies x_0 \in S\).
• (Topology)

Notes

• Union of infinitely many closed sets can be open.
• Intersection of infinitely many open sets can be closed.
• \(\displaystyle \{\}\) and \(\displaystyle \mathbb{R}\) are both open and closed

Compact

Compactness is a generalization of closed and bounded.

Definitions

• (Sequence) A set is sequentially compact if for every sequence from the set, there exists a subsequence which converges to a point in the set.
• (Topology) A set is compact if for every covering by infinitely many open sets, there exists a covering by a finite subset of the open sets.

Notes

• A set is sequentially compact iff it is closed and bounded

Continuity

Definitions of Continuity

The following definitions of Continuity are equivalent.

• (Order) A function \(\displaystyle f\) is continuous at \(\displaystyle x_0\) if for all \(\displaystyle \epsilon\) there exists \(\displaystyle \delta\) such that \(\displaystyle |x - x_0| \leq \delta \implies |f(x) - f(x_0)| \leq \epsilon\)
• (Sequences) A function \(\displaystyle f\) is continuous at \(\displaystyle x_0\) if \(\displaystyle \{x_n\} \rightarrow x_0 \implies \{f(x_n)\} \rightarrow f(x_0)\)
• (Topology) A function \(\displaystyle f\) is continuous at \(\displaystyle x_0\) if for all open sets \(\displaystyle V\) s.t. \(\displaystyle f(x_0) \in V\), \(\displaystyle f^{-1}(V)\) is an open set.
  • The preimage of an open set is open.
  • Continuous functions map compact sets to compact sets.