A topological space is a Set $X$, with a collection of distinguished Subsets called Open sets, called the topology of the set. These must satisfy:

• the union of an arbitrary collection of open sets is open;
• the intersection of any finite collection of open sets is open;
• The empty set $\emptyset$ and the whole set, $X$ itself are both open.

see definition of a topology. We can define a Basis of a topology that generates the topology in the same way that Open ball generate the Standard topology. most often we start with a basis and construct the topology. Given a Subbasis, which is just any collection of subsets covering X, the induced topology is just given by arbitrary unions of finite intersections

An equivalent definition is that a topological space is a Neighbourhood space $(X, \mathcal{N})$ in which, for all $x \in X$ and for all $N \in \mathcal{N}(x)$, there exists $N_1 \in \mathcal{N}(x)$ such that, for all $y \in N_1, N \in \mathcal{N}(y)$.

It can also be shown that: A neighbourhood space $(X, \mathcal{N})$ is a topological space if and only if each filter $\mathcal{N}(x)$ has a Filter base consisting of Open sets.

Remark: for family $C$ of subsets of a set $X$, there exists a unique 'smallest' topology on $X$ for which $C$ is a subbase: namely that topology whose open sets are defined to be all arbitrary unions of the collection of all finite intersections of elements of $C$.

Connections with lattices (comparing topologies)

The set of open sets in a topology forms a lattice, where the partial ordering is set inclusion (we say that one topology is finer than another or the latter is coarser than the former).

Also the set of topologies on a set $X$ can also be equipped with a natural lattice structure.

Analytical properties

In a topological space one can define fundamental notions of:

• convergence
• continuity

These are approached using neighbourhoods of a point, which are just open sets that contain that point. The family of neighbourhoods

Examples of topologies

Standard topology

Lower limit topology

Product topology

Discrete topology

Order topology

Subspace topology

some examples. Indiscrete topology (empty and whole set), discrete topology (all subsets of the set (Power set)), finite complement topology