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  1. Let $\{(X_i, T_i) : i = 1,2,\ldots, n\}$ be a collection of topological spaces and let $\sigma$ be a permutation of the symbols $1, 2,\ldots, n$. For $i=1,2,\ldots, n$ let $Y_i = X_{ \sigma (i)}$ and let $Y = Y_1 \times Y_2 \times \ldots\times Y_n$ and $X = X_1 \times X_2\times\ldots \times X_n $, each with product topology. Prove that $X$ and $Y$ are homeomorphic to each other.

  2. For any three spaces $X_1, X_2, X_3$ prove that $X_1\times(X_2 \times X_3)$ is homeomorphic to $(X_1\times X_2 )\times X_3$.


Can anyone tell me please how can I tackle these problems?

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In each of these problems you should be able immediately to write down the homeomorphism, and once you have that, proving that it is a homeomorphism is straightforward. Here’s a hint for (2): try the map $$h:X_1\times(X_2\times X_3)\to(X_1\times X_2)\times X_3:\big\langle x_1,\langle x_2,x_3\rangle\big\rangle\mapsto\big\langle\langle x_1,x_2\rangle,x_3\big\rangle\;.$$ –  Brian M. Scott Jun 26 '13 at 8:24
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extra hint to Brian's hint: a function $f: X \rightarrow Y \times Z$ is continuous iff $\pi_Y \circ f$ and $\pi_Z \circ f$ are continuous, where $\pi_A$ is the projection onto the factor $A$ in a product space. The same holds for more factors; in short, a map into a product is continuous iff all the compositions with the projections from that product are continuous. Use this repeatedly to get the continuity of all functions involved (and their inverses). –  Henno Brandsma Jun 26 '13 at 11:24
    
$\{\text{curly braces}\}$ in mathematical notation should be inside, not outside, the TeX code. Notice how this was done in my edit to the question. This assures proper alignment and spacing. Also, it makes possible things like this: $\left\{\begin{array}{c} A \\ B \\ C \end{array}\right\}$. –  Michael Hardy Jun 26 '13 at 12:57

1 Answer 1

There is a more general approach to this problem. If may already have learned about the general construction of the topological product of spaces $X_i$, indexed by a set $I$. The set $X:=\prod_I X_i$ consists of all sequences $(x_i)_{i\in I}$. The word "sequence" is a bit misleading, it does not restrict to countable sequences. Think of this element as a map from $I$ to $\bigsqcup_I X_i$ such that to each $i\in I$ we assign an element $x_i\in X_i$. The product comes with projections $p_j:\prod_I X_i\to X_j$. We then equip $X$ with the coarsest topology such that the projections are continuous, namely the topology generated by the subbase $\{p_i^{-1}(U_i)\mid U_i\textrm{ open in }X_i,\ i\in I\}$. Defined this way, $X$ has the following property:

For each family of continuous maps $(f_i:Y\to X_i)_{i\in I}$ there is exactly one map $f:Y\to X$ such that $p_i\circ f=f_i$. This map is continuous since the compositions $p_i\circ f$ are continuous.

It is an easy exercise to show that if $Z$ is another space with projections $q_j:Z\to X_j$ satisfying the same property, then $Z$ is isomorphic to $X$.

This may help you solve the problems. If you want to prove that a space is homeomorphic to the product, show that it satisfies this universal property. For the second problem, this will give you isomorphisms $X_1\times(X_2\times X_3) \cong X_1\times X_2\times X_3 \cong (X_1\times X_2)\times X_3$

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