Let f1 X1 rightarrow y1 f2 x2 rightarrow y2 be maps between

Let f_1 : X_1 rightarrow y_1, f_2: x_2 rightarrow y_2 be maps between topological spaces and let f = f_2Times f_2: x_1 Times x_2 rightarrow y_1 Times y_2 be the cartesian product of these two maps, defined by f((x_1, x_2)):=(f_1(x_1), f_2(x_2)), x_1 X_1, x_2 X_2. Show that F is continuous if and only if f_1 and f_2 are continuous.

Solution

Consider f to be continuous.

which implies as f₁ and f₂ are continuous, trivially.

Now suppose, f₁ and f₂ are continuous,

which imples, as x₁ tends to a, f₁(x₁) tends to f₁(a).

also,  as x₂ tends to b, f₂(x₂) tends to f₂(b).

Therefore as x₁ tends to \'a\' and as x₂ tends to \'b\'.

( f₁(x₁) , f₂(x₂) ) tends to ( f₁(a), f₂(b) ).

Therefore f is continuous.