Disjoint union (topology)

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In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other.

Contents

The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.

Definition

Let {Xi : iI} be a family of topological spaces indexed by I. Let

${\displaystyle X=\coprod _{i}X_{i}}$

be the disjoint union of the underlying sets. For each i in I, let

${\displaystyle \varphi _{i}:X_{i}\to X\,}$

be the canonical injection (defined by ${\displaystyle \varphi _{i}(x)=(x,i)}$). The disjoint union topology on X is defined as the finest topology on X for which all the canonical injections ${\displaystyle \varphi _{i}}$ are continuous (i.e.: it is the final topology on X induced by the canonical injections).

Explicitly, the disjoint union topology can be described as follows. A subset U of X is open in X if and only if its preimage ${\displaystyle \varphi _{i}^{-1}(U)}$ is open in Xi for each iI. Yet another formulation is that a subset V of X is open relative to X iff its intersection with Xi is open relative to Xi for each i.

Properties

The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: If Y is a topological space, and fi : XiY is a continuous map for each iI, then there exists precisely one continuous map f : XY such that the following set of diagrams commute:

This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map f : XY is continuous iff fi = f o φi is continuous for all i in I.

In addition to being continuous, the canonical injections φi : XiX are open and closed maps. It follows that the injections are topological embeddings so that each Xi may be canonically thought of as a subspace of X.

Examples

If each Xi is homeomorphic to a fixed space A, then the disjoint union X is homeomorphic to the product space A×I where I has the discrete topology.

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