Topology and Set Definitions

Definitions

Topology of a set:
A set that contains multiple subsets of that set

Open Sets:
Let's say that open sets O ₁ , O ₂ , O ₃ are all elements of the topology of x
O ₁ Union O ₂ has to also be an element of the Topology of X
Also, any finite number of intersetions of open sets are also an element of the topology of X
The null set is an element of the topology of x
The set itself is an element of the topology of x

Can also be defined by a set that is a neighborhood of all of its point
These are represented by a shape with dashed lines

Closed Sets
The basic definiton is the complement of a open set
This is represented by a closed line
EX: Imagine an open ball or circle with dashed lines, the complement would be the rest of the plane with a "hole" seperated with a solid line

Power set
All posible subsets
This is the biggest possible topology
This is not the usual topology of the plane

Open Neighborhood
Any open set that contains a point
An open neighberhood of a point (n) is an open set that contains the point
That open set is a neighborhood of every point inside of it Limit Points
A point is a limit point of a subset if every neighborhood of that point also contains a point of that subset.
This becomes useful when defining boundires since every point on the boundry of an open set, has a nieghborhood with points in that subset
This also means that a limit point does not have to be in the open set to have every neighborhood in it

Closure
All of the limit points of the set (ommited points, interior points, and the boundry)

Interior A point is in the interior if there is a neighborhood that lies entirely in the set
This means that boundry points are not the interior because no neighborhood exists where all of the points are entirely in the set

Different types of open sets

Open Ball(On the plane)
set of all points X where all points x where the distance between x and and the center of the ball is less than some real number
This means that it does not include the boundry of the "ball"
Called the base of the topology because you can create any element with unions of open balls

Open interval (On a line)
Still open sets
They could be not connected, and still be an open set

NOT OPEN SUBSETS
A point is not an open set becuase a finite intersection of "balls" woill never get you a point

Different Topologies

Usual Topology
Open Ball

Discrete Topology
Every Point is an open set
Work for every Set

Trivial Topology
Ony member of the set is the Null sets and Set itself
Work for every Set

These definitions are important when classifying sets and manifolds. We will continue to build on these terms in future posts.