On Sun, 04 May 2008 23:49:57 EDT, amu <amu786la@[EMAIL PROTECTED]
> wrote:
>is this right proof
Proof of what? It can't be a correct proof of the statement
"Let X,Y be topological spaces. Consider Z = X x Y and the product
topology genearted by the projections p_x, p_y. Let A be a subset of
Z.
Suppose that X,Y are locally path connected. Show that A is connected
if and only if A is path connected."
because that's _false_.
>Let C be a connected set that is also locally path connected. Pick any
point x in C, and let U be the set of points in C that are path connected
to x. Thus U is a subset of C.
>Let y be a point in U. Enclose y in an open set H in C, such that y is
path connected to all of H. Since an arc can run from x to y to anything
in H, H is in U. Therefore U is the union of open sets and is open,
relative to C.
>
>Let y be a point in C that is a limit point of U. Put an open set H
around y such that H is path connected. Let z be common to H and U. Now x
connects to z connects to y, and y is in U.
>
>Since U contains its limit points it is closed. thus U is both open and
closed in C. If U is not all of C, separate U and the rest of C in open
sets. This contradicts the fact that C is connected. Therefore U is all of
C, and C is path connected.
>
>In n dimensional space, every open ball is path connected, and every open
set is locally path connected, hence every open connected set is path
connected.
David C. Ullrich
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