# Topology of finite sets

## Nearness functions

### 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.

### 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.

## Creating topologies from sets

### The trivial topology

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

## 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.