---
tag: A486VR787Z
kind: definition
name: Disjoint union topology
msc: [54B99, 54B30]
defines: ZXJKSB30WF
uses: [0CQHFPGJ45, 02BH3MCBQB, 6ERJRRZK32, 0VGFHTM75Y]
sources:
  - url: https://en.wikipedia.org/wiki/Disjoint_union_(topology)
---

## Definition

Let $I$ be a set and let $(X_i)_{i \in I}$ be a family of [[topological spaces|0CQHFPGJ45]].
The disjoint union of the family is the set
$$X = \bigsqcup_{i \in I} X_i = \set{(i, x) : i \in I, \ x \in X_i}.$$
For each $i \in I$, the canonical injection $\iota_i \colon X_i \to X$ is $\iota_i(x) = (i, x)$.
The *disjoint union topology* on $X$ is the set of all $U \subseteq X$ such that $\iota_i^{-1}(U)$ is [[open|02BH3MCBQB]] in $X_i$ for every $i \in I$.

## Remarks

It is the [[final topology|6ERJRRZK32]] on $X$ for the family of maps $\iota_i$.
Each $\iota_i$ is an open [[topological embedding|00ZG9Y9TR4]], and the image of each $X_i$ is open and closed in $X$.
Universal property: a map $f \colon X \to Y$ to a topological space $Y$ is [[continuous|0VGFHTM75Y]] if and only if $f \circ \iota_i$ is continuous for every $i$.

Example: if each $X_i$ is a one-point space, then every subset of $X$ is open, so $X$ is the discrete space on $I$.
Indeed, the preimage of any subset under each $\iota_i$ is empty or a point, and both are open.

Non-example: let $X_0$ and $X_1$ be two copies of a two-point space with the trivial topology, and let $X = X_0 \sqcup X_1$.
The trivial topology on the four-point set $X$ is not the disjoint union topology.
The set $\iota_0(X_0)$ has open preimages in both pieces, but it is not open in the trivial topology.

Notation: $\bigsqcup_i X_i$ or $\coprod_i X_i$; for two spaces, $X_1 \sqcup X_2$.