there are different ways of defining the disjoint union. one way involves "tagging" X and Y that is:

X+Y = (Xx{1}) U (Yx{2}) (any singleton sets that are distinct could be used for the "tags").

if X,Y are already disjoint sets, then the disjoint union is simply XUY. but if X∩Y ≠ Ø, "tagging" the sets ensures that for an element a in X∩Y, we get two elements in X+Y:

namely (a,1) (from X) and (a,2) (from Y).

while, strictly speaking, X is not a subset of X+Y, we have the bijection:

x ↔ (x,1), so as SETS they are isomorphic.

the mapping j is the obvious one:

j(x) = x, if x is in X

j(y) = y, if y is in Y

perhaps an example will help:

let X = Y = R, the real number line. then R+R is two "separate" copies of the line (unlike say, RxR, which is a plane, or (Rx{0}) U ({0}xR), which is just the two coordinate axes). since a point in this space lies either on one line or the other, the image j(x) is just "whatever real number of the real line x belongs to is".

we might represent a point on the first line as (r,1), and a point on the second line as (s,2), if it happens that r = s, j maps both these points to r in RUR = R.

another, more abstract, way to characterize the disjoint union is as follows:

let X,Y be two topological spaces. then X+Y is a space with two continuous embeddings:

j_{1}:X→X+Y

j_{2}:Y→X+Y

such that if we have two continuous maps f:X→Z and g:Y→Z, there is a UNIQUE continuous map h:X+Y→Z with

hj_{1}= f

hj_{2}= g

this map h is often written f+g, each "component" keeps track of whether any z in the image of h came from X, or came from Y.

the point is that Xx{*}, where {*} is any singleton space, is naturally homemorphic to X:

f(x,*) = x and

g(x) = (x,*) are clearly both continuous (from a given topology on X, and the product topology on Xx{*}) (note that there is only ONE toplogy on {*} consisting of:

T = {Ø,*}. on a singleton space, ALL topologies (the indiscrete and the discrete, or whatever) are exactly the same, so in the product topology we have just two kinds of open sets:

UxØ = Ø, and Ux{*}, where U is open in X).

the idea is this:

suppose X = {Bob,Ted,Alice} and Y = {Bob,Carol,Ted}. let's form X' = {BOB,TED,ALICE} (which is really just the same set as X, we just capitalized everything).

then X+Y = {Bob,BOB,Ted,TED,ALICE,Carol}, and j is given by:

j(Bob) = Bob

j(BOB) = Bob

j(Ted) = Ted

j(TED) = Ted

j(ALICE) = Alice

j(Carol) = Carol

j "preserves" the elements of X or Y when they are unique, and "identifies" them if they lie in the intersection of X and Y.