topology continuity

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February 23rd 2009, 10:18 PM
tomboi03

topology continuity

Let F: X x Y -> Z. We say that F is continuous in each variable separately if for each y0 in Y, the map h: X-> Z defined by h(X)= F( x x y0) is continuous, and for each x0 in X, the map k: Y-> Z defined by k(y) =F(x0 x y) is continuous. Show that if F is continuous, then F is continuous in each variable separately.

I'm not sure how to do this....

if you guys would help me out that would be amazing!
Thank You
February 24th 2009, 07:57 PM
aliceinwonderland

Quote:

Originally Posted by tomboi03

Let F: X x Y -> Z. We say that F is continuous in each variable separately if for each y0 in Y, the map h: X-> Z defined by h(X)= F( x x y0) is continuous, and for each x0 in X, the map k: Y-> Z defined by k(y) =F(x0 x y) is continuous. Show that if F is continuous, then F is continuous in each variable separately.

Lemma 1. Let X, Y be topological spaces; let $f:X \rightarrow Y$ . Then the following are equivalent.
(i) f is continuous.
(ii) For each $x \in X$ and each neighborhood V of $f(x)$ , there is a neighborhood U of x such that $f(U) \subset V$ .

Let W be a neighborhood of $h(x)=F(x \times y_0)$ for any $x \in X$ and a fixed point $y_0 \in Y$ . Since F is continuous, there exists a neighborhood V of $x \times y_0$ such that $F(V) \subset W$ by lemma 1. Since $V$ is an open set containing $x \times y_0$ in $X \times Y$ , there exists an open set U in X containing x such that $U=p_X(V)$ , where $p_X:X \times Y \rightarrow X$ is an open projection map.

Now, for any neighborhood W of $h(x) = F(x \times y_0)$ , we have an open set $U=p_X(V)$ containing x such that $h(U)=F(U \times y_0) \subset F(V) \subset W$ .
Thus, h is a continous by lemma 1.

It is pretty similar to show that k is continuous.
February 25th 2009, 11:55 AM
benes

Quote:

Originally Posted by tomboi03

Let F: X x Y -> Z. We say that F is continuous in each variable separately if for each y0 in Y, the map h: X-> Z defined by h(X)= F( x x y0) is continuous, and for each x0 in X, the map k: Y-> Z defined by k(y) =F(x0 x y) is continuous. Show that if F is continuous, then F is continuous in each variable separately.

I'm not sure how to do this....

if you guys would help me out that would be amazing!
Thank You

Suppose F(x,y) is continuous at (a,b) then we must prove h(x) is continuous at a and k(y) is continuous at b.

If F(x,y) is continuous at (a,b) then given ε>0 there exists δ>0,such that

|F(x,y)-F(a,b)|<ε ,for all x,y satisfying |x-a|<δ,|y-b|<δ.

Since |b-b|=0<δ. By putting y=b in the above inequalities we have:

|F(x,b)-F(a,b)|<ε ,for all x satisfying |x-a|<δ.

But by definition F(x,b)=h(x),and F(a,b)=h(a), so the above inequalities become:

|h(x)-h(a)|<ε,for all x satisfying |x-a|<δ.

Hence h(x) is continuous at a.

In exactly the same way we prove k(y) is continuous at b.

The converse of the above theorem is not necessarily true