Product Spaces & Homeomorphism

**Turloughmack** · November 30th 2010, 01:15 AM

I have a question that I can't start. I think if I got the start of it I could finish it but I dont know what my first move would be

Let $X,T$ be the product space $\pi(X_i,T_i)$ . Given any 'origin' $\bar{x}=(\bar{x_i})_{i+1}$ in X and any $i_0 \in I$ .

Show that the map
$\mu: x_i_{0} \rightarrow X$ given by

$\mu(y) = (x_i)_{i+1}$ where $x_i = \bar{x_i}$ if $i$ is not equal to $i_0$ or $x_i = y$ if $i=i_0$

is a homeomorphism between $x_i_{0}$ and $\mu(x_i_{0})$

**tonio** · November 30th 2010, 06:19 AM

Originally Posted by Turloughmack

I have a question that I can't start. I think if I got the start of it I could finish it but I dont know what my first move would be

Let $X,T$ be the product space $\pi(X_i,T_i)$ . Given any 'origin' $\bar{x}=(\bar{x_i})_{i+1}$ in X and any $i_0 \in I$ .

Show that the map
$\mu: x_i_{0} \rightarrow X$ given by

$\mu(y) = (x_i)_{i+1}$ where $x_i = \bar{x_i}$ if $i$ is not equal to $i_0$ or $x_i = y$ if $i=i_0$

is a homeomorphism between $x_i_{0}$ and $\mu(x_i_{0})$

Perhaps it's only me, but I can't make any sense of your post: you introduce us to X,T (presumably, X is

a points set and T is a topology on it), then you write $\pi(X_i,T_i)$ , which I guess (guess!) means the product of

top. spaces $(X_i,T_i)$ , though the little pi is the usual notation, within this context, for the fundamental group.

Thus, in fact it seems to be that $X=\prod\limits_{i\in I}X_i$ , whereas T is the heaven knows what topology on this
(The prod. topology? The box topology? Other topology?) . You then take an "origin" $\bar{x}=(\bar{x_i})_{i+1}\in X\,,\,\,i_0\in I$ ...why

the index is i+1?? And what or who is $i_0$ ? The set $I$ is, I presume, the original one indexing the product above...?

Then you define a map from $x_{i_0}$ (What is this? A coordinate from the element $(x_i)$ above, or what?) to $X$ , but

you define it on some $y$ ....and then you ask to show that $x_{i_0}$ is homeom. to its image...?

Please try to be way clearer or, preferably, post a link to the original question...or dismiss this post

if I made a whole mess from something simple.

Tonio

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