Embeddings of X in Y and Y in X Defined by f(x) and g(y)

[tomboi03]

x₀ $$\in$$X and y₀$$\in$$Y,
f:X$$\rightarrow$$X x Y and g: Y$$\rightarrow$$X x Y defined by
f(x)= x x y₀ and g(y)=x₀ x y are embeddings

This is all I have...
f(x): {(x,y): x$$\in$$X and y$$\in$$Y}
g(y): {(x,y): x$$\in$$X and y$$\in$$Y}

right?
soo... embeddings are... one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.i don't know how to do go about this...

 

[CompuChip]

An embedding is literally what it says: a function that embeds one topological space in another. For example, if n > 1 then
$$f: \mathbb{R} \to \mathbb{R}^n, x \mapsto (x, 0, 0, \cdots, 0)$$
embeds the real line in some space (like the plane or three-dimensional Euclidean space),
$$g: S^1 \to \mathbb{R}^2, \theta \mapsto (\cos\theta, \sin\theta)$$
embeds the circle in the plane, but you can also embed it in a sphere
$$h: S^1 \to S^2, \theta \mapsto (\cos\theta, \sin\theta, 0)$$
which maps the circle S^1 to the equator of the sphere S^2 (you can even compose this with any rotation, which just maps it to some other great circle on the sphere).

The function you gave, for example
$$X \to X \times Y, x \mapsto (x, y_0)$$
is a simple example of a general embedding. Basically what you do is choose a fixed point in Y, and then embed X in X x Y by simply filling in the "missing" coordinates with y0. For example, the embedding of the real line in Euclidean space which I called f above, can be obtained in this way: set $X = \mathbb{R}, Y = \mathbb{R}^{n - 1}, y_0 = \vec 0_{n-1}$ where $\vec 0_{n-1}$ is the zero vector in $\mathbb{R}^{n-1}$.