Geometrical interpretation of disjoint union

Question

Can anybody make me understand the geometrical interpretation of the concept $Disjoint$ $Union $? Mathematically it's fine but I'm unable to grasp it geometrically.

Answer 1

Let us take the case of a topological space $X$ with two nonempty subspaces $A,B$ whose union is $X$. In general the topologies of $A,B$ do not determine that of $X$. However if $A \cap B$ is empty and a set $U$ is open in $X$ if and only if $U$ intersects $A,B$ in sets open in $X$, then intuitively $X$ is "in two bits" and we write $X =A \sqcup B$, the disjoint union of $A,B$. This can also be usefully described by saying that the function $f: X \to \{0,1\}$ taking $A$ to $0$ and $B$ to $1$, where $\{0,1\}$ has the discrete topology, is continuous.

So we say $X$ is connected if any continuous function from $X$ to the discrete space $\{0,1\}$ is constant.This characterisation is often quite useful.

I leave you to generalise to $n$ subsets, or any "disjoint union", and to draw examples.