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

\(X\in T\)

\(\varnothing \in T\)

Unions of \(T\) are in \(T\)

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

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

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

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 \(M\subseteq N\)

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

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

\(U\) is an open set if it is a neighbourhood for all its 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 \((0,1)\).

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 \(1\).

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 \((0,1)\).

The complement of any open set is a closed set.

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

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

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

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

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

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

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.

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

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.

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

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

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

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.