Let $X$ and $Y$ be reasonable spaces. Since $\mathbb{R}^{\infty}$ is contractible, 
$$
X \times \mathbb{R}^{\infty} \cong Y \times \mathbb{R}^{\infty} \;\;\; \implies \;\;\; X \simeq Y.
$$


Is the converse also true?


My vague intuition: the factors of $\mathbb{R}^{\infty}$ provide so much extra room that there will never be a geometric obstruction to producing a homeomorphism. Evidently, there is no homotopy-theoretic obstruction, so maybe the converse is true.


On the other hand, I really have no idea and could be missing something basic. For example, the plane with two punctures is homotopy equivalent to a wedge of two circles. However, I do not know about a homeomorphism
$$
(\mathbb{C} - \{0, 1\}) \times \mathbb{R}^{\infty} \overset{?}{\cong} (S^1 \vee S^1) \times \mathbb{R}^{\infty}.
$$




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**Clarification about the meaning of $\mathbb{R}^{\infty}$ and the intent of the question**


When I wrote the question, I had in mind the infinite union $\cup\_n \mathbb{R}^n$ inside the product $\mathbb{R}^{\mathbb{N}}$. However, since I would like the answer to the question to be "yes," I am also interested in other versions of $\mathbb{R}^{\infty}$.


I asked the question because of an algebraic limiting construction in a paper I'm writing, and I felt that a topological version of the limit would be satisfying. The algebraic version is already working for spaces like $X \times \mathbb{C}^n$ for large-enough $n$, and converges algebraically to some limiting group, but this doesn't give too many hints about the topology I should use on $\mathbb{C}^{\infty}$, or if the limit can even be considered topologically.


My application involves singular homology, and the direct limit topology is well-suited to this application, but other choices may be as well.



If $X$ and $Y$ are homotopy equivalent, then are $X \times \mathbb{R}^{\infty}$ and $Y \times \mathbb{R}^{\infty}$ homeomorphic?
This question is answered by two classical theorems of infinite-dimensional topology, which can be found in the books of [Bessaga and Pelczynski](https://books.google.com.ua/books/about/Selected_Topics_in_Infinite_Dimensional.html?id=7n9sAAAAMAAJ&redir_esc=y), [Chigogidze](https://books.google.com.ua/books?id=uaIrAAAAYAAJ&q=Chigogidze&dq=Chigogidze&hl=uk&sa=X&ved=0ahUKEwj65v698rPZAhWG2qQKHXL9Cc0Q6AEIJzAA) or [Sakai](https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxrc2FrYWlpZHRvcG9sb2d5fGd4OjU1ZmE2MTY1ZWZhNzg0MTY).


**Factor Theorem.** For any Polish absolute neighborhood retract $X$ (= neighborhood retract of $\mathbb R^\omega$) the product $X\times\mathbb R^\omega$ is an $\ell\_2$-manifold.


**Classification Theorem.** Two $\ell\_2$-manifolds are homeomorphic if and only if they are homotopy equivalent.


So, the *reasonable space* in your question should read as *a Polish absolute neighborhood retract*. By the way, this class of spaces includes all (countable locally) finite simplicial complexes, mentioned in the answer of Igor Rivin.




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**Remark on possible generalizations.** Chapter IX of the book of [Bessaga and Pelczynski](https://books.google.com.ua/books/about/Selected_Topics_in_Infinite_Dimensional.html?id=7n9sAAAAMAAJ&redir_esc=y) contains generalizations of the above two theorems to manifolds modeled on normed spaces $E$, which are homeomorphic to $E^\omega$ or $E^{\omega}\_0:=\{(x\_n)\_{n\in\omega}\in E^\omega:\exists n\in\omega\;\forall k\ge n\;(x\_k=0)\}$.


Chapter 7 of [Chigogidze's book](https://books.google.com.ua/books?id=uaIrAAAAYAAJ&q=Chigogidze&dq=Chigogidze&hl=uk&sa=X&ved=0ahUKEwj65v698rPZAhWG2qQKHXL9Cc0Q6AEIJzAA) contains generalizations of the Factor and Classification Theorems to manifolds modeled on uncountable products of lines.


Chapter 5 of [Sakai's book](https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxrc2FrYWlpZHRvcG9sb2d5fGd4OjU1ZmE2MTY1ZWZhNzg0MTY) contains generalizations of the Factor and Classification Theorems to manifolds modeled on the direct limits $\mathbb R^\infty$ and $Q^\infty$ of Euclidean spaces and Hilbert cubes, respectively.


So, typical generalizations of Factor and Classification Theorems look as follows:



> 
> **Theorem.** Let $E$ be a reasonable model space (usually it is an infinite-dimensional locally convex space with some additional properties).
> 
> 
> $\bullet$ For any neighborhood retract $X$ of $E$ the product $X\times E$ is an $E$-manifold.
> 
> 
> $\bullet$ Two $E$-manifolds are homeomorphic if and only if they are homotopically equivalent.
> 
> 
> 




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Depending on the meaning of your model space $\mathbb R^\infty$ the meaning of a *reasonable space* also changes. If $\mathbb R^\infty$ is the direct limit of Euclidean spaces, then a *reasonable space* means an absolute neighborhood extensors which is a direct limits of finite dimensional compacta. If $\mathbb R^\infty$ is the union of the Euclidean spaces in the countable product of lines, then a *reasonable space* is a locally contractible space which is the countable union of finite-dimensional compact subsets.


