Definition:Topology

Definition
Let $X$ be a set.

A topology on $X$ is a subset $\vartheta \subseteq \mathcal P \left({X}\right)$ of the power set of $X$ that satisfies the following axioms:


 * $(1): \quad$ The union of an arbitrary subset of $\vartheta$ is an element of $\vartheta$.
 * $(2): \quad$ The intersection of any two elements of $\vartheta$ is an element of $\vartheta$.
 * $(3): \quad X$ is an element of $\vartheta$.

If $\vartheta$ is a topology on $X$, then $\left({X, \vartheta}\right)$ is called a topological space.

The elements of $\vartheta$ are called the open sets of $\left({X, \vartheta}\right)$.

Elementary Properties

 * In General Intersection Property of Topological Space, it is proved that a topology can equivalently be defined by the properties:


 * $(1): \quad$ The union of an arbitrary subset of $\vartheta$ is an element of $\vartheta$.
 * $(2): \quad$ The intersection of any finite subset of $\vartheta$ is an element of $\vartheta$.


 * In Empty Set is Element of Topology it is shown that in any topological space $\left({X, \vartheta}\right)$ it is always the case that $\varnothing \in \vartheta$.

Also see

 * $\sigma$-algebra, which looks similar on the surface to a topology, but closed (in the algebraic sense) under countable unions. A topology has no such limitation on countability.