---
tag: 2CG46QFBHW
kind: definition
name: Vietoris topology
msc: [54B20]
defines: BJVRFFJTV6
uses: [0CQHFPGJ45, 0ZCK8DQYN1, 02BH3MCBQB, 09S8DV34Y0, 0R6EWFKRCF]
sources:
  - url: https://en.wikipedia.org/wiki/Hypertopology
---

## Definition

Let $X$ be a [[topological space|0CQHFPGJ45]], and let $\mathrm{CL}(X)$ be the set of nonempty [[closed|0ZCK8DQYN1]] subsets of $X$.
For an [[open|02BH3MCBQB]] set $U \subseteq X$, put
$$U^{+} = \set{A \in \mathrm{CL}(X) : A \subseteq U}, \qquad U^{-} = \set{A \in \mathrm{CL}(X) : A \cap U \ne \varnothing}.$$
The *Vietoris topology* on $\mathrm{CL}(X)$ is the topology generated by the [[subbasis|09S8DV34Y0]] of all sets $U^{+}$ and $U^{-}$ with $U$ open in $X$.

## Remarks

The same subbasis, intersected with a smaller family, defines the Vietoris topology on that family, for example on the nonempty [[compact|0R6EWFKRCF]] subsets $\mathrm{K}(X)$.
If $X$ is a metric space, the Vietoris topology on $\mathrm{K}(X)$ is the topology of the Hausdorff metric.
If $X$ is compact Hausdorff, then $\mathrm{CL}(X)$ with the Vietoris topology is compact Hausdorff.
If $X$ is $T_1$, the map $x \mapsto \set{x}$ is a [[topological embedding|00ZG9Y9TR4]] of $X$ into $\mathrm{CL}(X)$.

Example: let $X = [0, 1]$, let $A = [0, 1]$, and let $A_n = \set{k/n : 0 \le k \le n}$.
Each basic neighbourhood of $A$ has the form $U^+ \cap V_1^- \cap \cdots \cap V_m^-$ with $A \subseteq U$ and each $V_i$ open and nonempty.
For large $n$, the set $A_n$ meets each $V_i$ and lies in $U = X$, so $A_n \to A$.

Non-example: the sets $\set{0}$ do not converge to $[0, 1]$, though they lie in every $U^+$ with $[0, 1] \subseteq U$.
The open set $V = (1/2, 1]$ gives a neighbourhood $V^-$ of $[0, 1]$ that does not contain $\set{0}$.

The resulting space is called a hyperspace of $X$, and the topology is also called the finite topology or exponential topology.