Given a set , a pair of sets is called a topological space if is a family/ collection of subsets of that satisfies the following properties:
and .
The union of any collection of sets in is also in .
The intersection of any finite collection of sets in is also in .
If , then
in which case, we say that is a topology on , and the sets in are called open sets of the topological space .
Euclid Space as a Topological Space
Euclidean Metric
The Euclid space can be regarded as a topological space by defining the topology using open balls, which is defined using the Euclidean metric on .
Given two points in , say and , the Euclidean metric is defined as
Note that this makes a metric space, which is a special type of topological space, where distance between points is explicitly defined.
Open Balls
Now, given a point and a positive real number , we can define the open ball centered at with radius as:
which is the set of all points in that are within a distance from the point .
Open Sets in
Using open balls, we can define an open set in as follows:
A set is an open set, if
Important Notions
Closed Sets
A set is called a closed set if its complement in , denoted by , is an open set:
Neighborhoods
Given a topological space , A set is called a neighborhood of a point if
where is the interior of .
Note that if neighborhood of is an open set, then is called an open neighborhood of . If neighborhood of is a closed set, then is called a closed neighborhood of .
Base/ Basis of a Topology
Given a topological space , a sub-family is called an open base of the topology , if
Or equivalently, if every open set in can be expressed as a union of sets in :
Note that there could be more than one base for a given topological space .
However, if one of them happens to be a countable set, then we say that the topological space is second-countable.
Example: Euclidean Space as a Topological Space
Given a point in an open set , we can always find an open ball centered at with radius such that (by definition and any open set in contain a rational point).
Then, fix a point ().
Also, take any point .
From the triangle inequality, we have
which means that , and thus .
In summary,
where
This means that is a countable base for , and thus is a second-countable topological space.
Hausdoff space
A topological space is called a Hausdorff space , if for any two distinct points , there exists two disjoint open sets such that and .
We say that these two points are separated by the two open sets and .
A regular Hausdorff space is for any closed sets , there exists two disjoint open sets such that and .
Again, notice that Euclidean space is a Hausdorff space, since for any two distinct points , we can always find two disjoint open balls and with radius such that and .
Maps between Topological Spaces
Continuous Maps
Given two topological spaces and , a map is called continuous if
Equivalently, we can define continuity in terms of neighborhoods:
This reduces to the usual definition of continuity for at a point :
This is because is equivalent to .
By extension,
which satisfies the definition of continuity in terms of neighborhoods.
Properties of Continuous Maps
Composition of continous maps is continous.
Given three topological spaces , and , and continous maps between them, and , the composition of and , denoted by , is also continuous:
Homeomorphisms
Given two topological spaces and , a map is called a homeomorphism if:
is bijective
is continuous
its inverse map is also continuous.
In this case, we say that and are homeomorphic, and denote as:
Now, notice that if is a homeomorphism, is also a homeomorphism.
Similarly to the composition of continuous maps, the composition of homeomorphisms is also a homeomorphism.