Diagonal Map

Definition

A diagonal map is a map from

$$\Delta :X\to X\times Xx\to (x,x)$$

which is also continuous.

Two Spaces

In general, for any metric spaces $X_1,X_2,Y_1,Y_2$,

$$f_1:X_1\to Y_1{f}_2:X_2\to Y_2$$

are continuous, then

$$f_1\times f_2\to X_1\times X_2\to Y_1\times Y_2$$

are continuous.

Generalization

In general,

$$f_1\times \cdots \times f_n:X_1\times \cdots \times X_n\to Y_1\times \cdots \times Y_n$$

is continuous.