Topology basics

Sets #

2. Interval, Neighborhood, Interior, 3. Limit Point, Derived Set, Closure, 3.5 Compact, Dense, Perfect, Connected Sets

Open Set, Closed Set, Bounded Set, Compact Set, Connected Set: Topology part-3 - YouTube

Boundary Point #

A point P is boundary point of domain D if every neighborhood of P contains a point in D and a point in D'.

Connected Set #

In the context of topology

A set S is said to be connected if any two points in it can be joined by path of segments whose points are in S.

Domain #

An open connected set is called a domain

Closed Domain #

domain + boundary points = closed domain

continuous 2 var function #

$$ \forall \epsilon>0 (\exists \delta (|f(x,y)-f(x_{0},y_{0})|< \epsilon)) \forall |x-x_{0}|< \delta, |y-y_{0}|<\delta $$

bounded func #

Let f be a function be defined on a domain or closed domain D, the function is bound on D if

$$\exists M(\forall x,y(|f(x,y)|<M))$$