# Thread: Product Spaces & Homeomorphism

1. ## Product Spaces & Homeomorphism

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

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

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

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

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

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

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

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

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 $\displaystyle \pi(X_i,T_i)$ , which I guess (guess!) means the product of

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

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

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

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

you define it on some $\displaystyle y$....and then you ask to show that $\displaystyle x_{i_0}$ 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