topological space

A topological space is a set $X$ together with a set $𝒯$ whose elements are subsets of $X$, such that

• $\mathrm{\varnothing }\in 𝒯$

• $X\in 𝒯$

• If $U_j\in 𝒯$ for all $j\in J$, then ${\bigcup }_{j\in J}{U}_j\in 𝒯$

• If $U\in 𝒯$ and $V\in 𝒯$, then $U\cap V\in 𝒯$

Elements of $𝒯$ are called open sets of $X$. The set $𝒯$ is called a topology on $X$. A subset $C\subset X$ is called a closed set if the complement $X\setminus C$ is an open set.

A topology ${𝒯}^{\prime }$ is said to be finer (respectively, coarser) than $𝒯$ if ${𝒯}^{\prime }\supset 𝒯$ (respectively, ${𝒯}^{\prime }\subset 𝒯$).

Examples

• The discrete topology is the topology $𝒯=𝒫(X)$ on $X$, where $𝒫(X)$ denotes the power set of $X$. This is the largest, or finest, possible topology on $X$.

• The indiscrete topology is the topology $𝒯=\{\mathrm{\varnothing },X\}$. It is the smallest or coarsest possible topology on $X$.

• Subspace topology

• Product topology

• Metric topology