A topological space is defined by a set of axioms that apply to the _set of [open](open%20set) subsets_, $\mathscr{O}$. Knowing that the complement of an open set is [closed](closed%20set) the topological space is again defined for the _set of closed sets_ $\mathscr{A}$ in terms of axioms that apply to the _set of closed sets_. It is also true that _the complement of a closed set is open_. Thus we may also define a topological space using the axioms for a _set of closed subsets_ and say that a topological space is also defined for _set of open sets_ whose elements the complements of the elements of the _set of closed sets_. Therefore, defining topological spaces with an open set to start with is a matter of convention. There are additional definitions in terms of axioms provided by [Haussdorff](Topological%20spaces.md#As%20originally%20defined%20by%20Haussdorff) and [Kuratowski](Topological%20spaces.md#As%20defined%20by%20Kuratowski) that are equivalent. # Topological groups  ([... see more](Topological%20group.md)) # Open set definition of topological space A _topological [[Spaces]]_ is a pair, $(X,\mathscr{O})$, containing a set $X$ and a _set of subsets of X_ $\mathscr{O}.$ The subsets $\mathscr{O}$ are called _[open](open%20set) subsets._ The following axioms hold for $Y,Z\in\mathscr{O}$: 1. A union between open sets, $Y\cup Z$, is also open. 2. An intersection between any two open sets $Y \cap Z$ is also open. 3. The empty set, $\emptyset$, and $X$ are both open. # Closed set definition of topological space Consider the closed set, $\mathscr{A}=\{X \setminus V|V\in\mathscr{O}\}$, a topological space is defined with pair $(X,\mathscr{A})$ for $Y,Z\in\mathscr{A}.$ 1. A union between closed sets, $Y\cup Z$, is also closed. 2. An intersection between any two closed sets $Y \cap Z$ is also closed. 3. $X$ and $\emptyset$ are both also closed. # Topological spaces as defined by Haussdorff # Topological spaces as defined by Kuratowski #MathematicalFoundations/Geometry/Topology