๐Ÿ—๏ธ ฮ˜ฯฯตฮทฮ ฮฑฯ„ฯ€๐Ÿšง (under construction)

Topology

Topology on a set X

A topology on X is a collection ๐’ฏ of subsets of X, that is: it is a subset of the P(X), with the following properties:

  • โˆ…,Xโˆˆ๐’ฏ
  • For any AโŠ†T
  • โ‹ƒAโˆˆT
  • For any CโŠ†T such that C is finite
  • โ‹‚CโˆˆT
The set X along with ๐’ฏ satsifying the above conditions is called a topological space and is denoted by (X,๐’ฏ)
Open set
Suppose (X,๐’ฏ) is a topological space, if Uโˆˆ๐’ฏ then we say that U is open with respect to X.
a set filled with open sets is open
Let X be a topological space, and AโŠ†X. Suppose that for each xโˆˆA there is an open set U containing x such that UโŠ†A. Show that A is open in X.
The Finite Complement Topology
Let X be a set and define the set FC={UโŠ†X:|XโงตU|<โˆž}โˆช{โˆ…}, then FC is a topology and we denote it by TFC
The Countable Complement Topology
Let X be a set and define the set CC={UโŠ†X:|XโงตU|โ‰คโ„ต0}โˆช{โˆ…} where we've used โ„ต0 then CC is a topology and we denote it by TC

Note that ๐’ฏ={X}โˆช{UโŠ‚X:Xโˆ’U is infinite}. is not a topology so long as X is infinite. For example pick some pโˆˆX then every singleton {q} where qโ‰ p is open in X because Xโงต{q} is infinite, if it were to be a topology then we would know that โ‹ƒqโ‰ p{q}=Xโงต{q} is open, but Xโงต(Xโงต{q})={q} which is finite, thus a contradiction, so it cannot be a topology.

The Intersection of Topologies Is a Topology
If ๐”— is a collection of topologies then โ‹‚๐”— is a topology.
The Intersection of a Collection of Sets That Are Supersets of a Given Set and Satisfy a Property Is the Smallest Set Which Satisfies the Property and Is Still a Superset of the Given Set
Let X be a set and Q a predicate then suppose that C is a collection of sets such that for any CโˆˆC we have XโŠ†C and Q(C) then if Q(โ‹‚C) holds true then it is the smallest such set where it holds true
Given a Family of Topologies There Is a Unique Smallest Topology Containing All of Them
Suppose that T is a collection of topologies on X, and let F be the collection of topologies such that for any FโˆˆF, we know that for every TโˆˆT we have TโŠ†F, then โ‹‚F is the unique smallest topology on X that contains all the topologies in T
Given a Family of Topologies There Is a Unique Largest Topology Contained in Every Topology in the Family
Suppose that T is a collection of topologies on, then โ‹‚T is the unique largest topology that is contained in T for each TโˆˆT.
Example of the Largest and Smallest Topology
With X={a,b,c}, T1={โˆ…,X,{a},{a,b}} and T2={โˆ…,X,{a},{a,b}}, find the smallest topology containing both, and the largest one contained in both.
The Power Set is a Topology
Given a set X, then P(X) is a topology on it.
finer and coarser topologies

suppose that ๐’ฏ and ๐’ฏโ€ฒ are two topologies on a given set X. If ๐’ฏโŠ†๐’ฏโ€ฒ, then ๐’ฏโ€ฒ is finer than ๐’ฏ. If the reverse inclusion is true, then we say that ๐’ฏโ€ฒ is coarser than ๐’ฏ, there are also strict variations of these definitions for the strict inclusions.

comparable topologies

given two topologies, they are comparable if at least one is finer than the other