Topology of finite sets

Nearness functions

Topologies

Topologies on sets

\(T\) is a topology on set \(X\) if:

  • \(X\in T\)

  • \(\varnothing \in T\)

  • Unions of \(T\) are in \(T\)

  • Intersections of \(T\) are in \(T\)

Examples of topologies: The trivial topology

The trivial topology contains only the underlying set and the empty set.

Examples of topologies: The discrete toplogy

The discrete toplogy contains all subsets of the underlying set (is this the power set?)

Neighbourhoods

Neighbourhood topology

We have a set \(X\).

For each element \(x\in X\), there is a non-empty set of neighbourhoods \(N\in \mathbf N(x)\) where \(x\in N\subseteq X\) such that:

  • If \(N\) is a subset of \(M\), \(M\) is a neighbourhood.

  • The intersection of two neighbourhoods of \(x\) is a neighbourhood of \(x\).

  • \(N\) is a neighbourhood for each point in some \(M\subseteq N\)

Topological distinguishability

If two points have the same neighbourhoods then they are topologically indistinguishable.

For example in the trivial topology, all points are topologically indistinguishable.

Open sets

\(U\) is an open set if it is a neighbourhood for all its points.

Open and closed sets

Limit points and closure

Limit points

A point \(x\) in the topological set \(X\) is a limit point for \(S\subset X\) if every neighbourhood of \(x\) contains another point in \(S\).

For example \(-1\) is a limit point for the real numbers where \(S\) is \([0,1]\) (or \((0,1)\).

Closure

The closure of a subset of a topological space is the subset itself along with all limit points.

So the closure of \(|x|<1\) includes \(-1\) and \(1\).

Boundries and interiors

The boundry of the subset \(S\) of a topology is the intersection with the closure of \(S\) with the closure of the complement of \(S\).

So the boundry of both \((0,1)\) and \([0,1]\) are \(0\) and \(1\).

The interior of \(S\) is \(S\) without the boundry.

So the interior of \((0,1)\) and \([0,1]\) are both \((0,1)\).

Closed sets

The complement of any open set is a closed set.

A set can be open, closed, both or neither.

Compactness

Covers

A space \(X\) is covered by a set of subsets of \(X\), \(C\), if the union of \(C\) is \(X\).

Subcover

A subset of \(C\) which still covers \(X\) is a a subcover.

Open cover

\(C\) is an open cover if each member is an open set.

Universal cover

Bases of topologies

Subset \(B\) of topology \(T\) is a base for \(T\) if all elements of \(T\) are unions of members of \(B\).

Second-countable space

If \(B\) is finite then the toplogy is a second-countable space.

Separation

Connected and separated sets

Two subsets of \(X\) in topological space \(T\) are separated if each subset is disjoint from the other’s closure.

So \([-1,0)\) and \((0,1)\) are separated.

\([-1,0]\) and \((0,1)\) are not separated.

Sets which are not separated are connected.

Cartesian products

Box topology

Product topology

Creating topologies from sets

The trivial topology

A topology which contains just \(X\) and \(\varnothing \) is the trivial topology.

Discrete topology

Taxonomy of spaces

Lindelöf space

In a Lindelöf space all open covers have countable subcovers.

This is weaker than compactness, which requires that every open cover has a finite subcover.

Kolmogorov space

In a Kolmogorov (or \(T_0\)) space, for every pair of points there is a neighbourhood containing one but not the other.

Local properties

Local properties

Locally, a topology may have properties which are not present globally.

Locally compact spaces

Locally connected spaces

TO INF

Hausdorf space

In a Hausdorf (or T\(_2\)) space, any two different points have neighbourhoods which are disjoint.

Compact spaces

A space \(X\) is compact if each open cover has a finite subcover.

If we can define a cover which does not have a finite subcover, then the space is not compact.

For example an infinite cover could be tend towards \((0,1)\), eg as \(\dfrac{1}{n},1-\dfrac{1}{n}\)

This covers \((0,1)\), but there is no finite subcover. As a result \((0,1)\) is not compact.