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I am looking for a bunch of examples of homeomorphisms in the finite product topology.

I need it for a proof

Someone mentioned my question was vague .

The theorem is:

Let n>1 and let $X_1,...,X_n$ be topo. spaces Then each $X_i $is homeomorphic to a subspace of the product space

I asked for different functions after seeing this post:

Homeomorphism in product and box topology

Can the idea in this post be used to prove the theorem I am trying to prove.

I saw a proof of it in Topology Without Tears .So I did one version based on it

I want to do a different one. That is why I am asking.

The more interesting the better Thanks

1 Answers1

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Not sure if this is what you're looking for, but here are some examples:

(1) Consider $n \in \mathbb{Z}$ with prime factorization $n=\varepsilon p_1^{v_1}\dots p_k^{v_k}$, where $\varepsilon \in \{1,-1\}$ and $v_i \in \mathbb{N}$. By the chinese remainder theorem, we have a ring isomorphism $$ \Phi : \mathbb{Z}/n \stackrel{\sim}{\longrightarrow} \prod_{i=1}^k \mathbb{Z}/(p_i^{v_i}). $$ We can, however, consider the quotient topology $\tau$ on $\mathbb{Z}/n$, induced by the canonical projection $\pi : \mathbb{Z} \twoheadrightarrow \mathbb{Z}/n$. We see that $(\mathbb{Z}/n,\tau)$ is a discrete topological space, because $\mathbb{Z}$ is discrete. Moreover, the space $\prod_{i=1}^k \mathbb{Z}/(p_i^{v_i})$ is discrete as a product of discrete spaces. It follows that $\Phi$ is a homeomorphism.

(2) Let $f:X\to Y$ me a map between topological spaces. Denote with $$ \Gamma(f) := \{ (x,f(x)) \mid x \in X \} \subset X \times Y $$ the graph of $f$, equipped with the subspace topology of the product topology on $X \times Y$. Let $p_X : X \times Y \to X$, $(x,y) \mapsto x$ denote the projection in the first component. Then, it can be shown that $$ p_X|_{\Gamma(f)} : \Gamma(f) \to X, \qquad (x,y) \mapsto x $$ is a homeomorphism if and only if $f$ is continuous.

I hope this was helpful!

jasnee
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