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**Amer** I think I can define the projection function from $\displaystyle P_i:\left(\prod_{k=0}^{n} X_k,T_p\right) \cong \left(X_i,T_i\right)$

this is one-one and onto and the projection is continuous and the inverse for the projection is continuous thus it is homeomorphism right

$\displaystyle P_i$ is onto and continuous but it not necessarily one-to-one. For example let $\displaystyle \begin{aligned}P_x : \mathbb{R}\times \mathbb{R}& \to \mathbb{R}\\ (x,y)& \mapsto x \end{aligned}.$

We have $\displaystyle P_x((1,0))=P_x((1,2))=1$ so $\displaystyle P_x$ is not injective.

However the restriction of $\displaystyle P_i$ to the set $\displaystyle Y_i=\{a_1\}\times\cdots\times\{a_{i-1}\}\times X_i\times \{a_{i+1}\}\times \cdots\times \{a_n\}$ that I defined in my previous post is an homeomorphism... it is the function I named $\displaystyle \pi$.