Topological Space

In topology, a topological space consists of a set \(X\) together with a "topology" \(\mathcal{T}\), which is a set of subsets of \(X\) subject to a few conditions:

1. \(\mathcal{T}\) must have both the empty set and \(X\) itself as elements.
2. The union of any number of elements of \(\mathcal{T}\) must be an element of \(\mathcal{T}\).
3. The intersection of any finite number of elements of \(\mathcal{T}\) must be an element of \(\mathcal{T}\).

The elements of \(\mathcal{T}\) are called the open sets or open subsets of the topological space.

Examples of Topological Spaces

The simplest nontrivial topological space is the Sierpinski Space.

Category Theory

In Category Theory, the category \(\text{Top}\) has as objects all possible topological spaces, with [[Homeomorphism|homeomorphism]] as morphisms. There exists a functor from this category to the category of sets which takes a topological space to its topology.