# Advanced Calculus

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Advanced Calculus as taught in Fitzpatrick's book.

## Contents

## Sequences

## Topology

### Closed

The folllowing definitions of Closed Sets are equivalent.

- (Order)
- (Sequences) A set is clsoed if it contains all its limit points. That is , .
- (Topology)

- Notes

- Union of infinitely many closed sets can be open.
- Intersection of infinitely many open sets can be closed.
- and 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

### Metric Space

## Continuity

### Definitions of Continuity

The following definitions of Continuity are equivalent.

- (Order) A function is continuous at if for all there exists such that
- (Sequences) A function is continuous at if
- (Topology) A function is continuous at if for all open sets s.t. , is an open set.
- The preimage of an open set is open.
- Continuous functions map compact sets to compact sets.