# Questions tagged [product-space]

For questions about the structure of product space, in the context of topology (including metric and normed spaces) or measure theory. Use other tags to indicate the context.

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### What is this topology?

I'm working through a text that is using some topology. It defines the following topology, I'm confused on what it would look like. In topology, $2^\kappa$ denotes $^\kappa\{0,1\}$, where $2= \{0,1\}$...
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### homeomorphicity of 2 subspaces in $\mathbb R^2$

I am looking at the $2$ subspaces $X=L\cup(\{0\}\times\mathbb N)$ and $Y=L\cup(\{0\}\times\mathbb Z)$ where $L:=\{(\frac{1}{n},y):n\in \mathbb N, y\in \mathbb R\}$. I wondering if these $2$ spaces are ...
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### Is the Projection of a Partition of a Product Space always Disjoint in at least one Factor Space?

I came across this while attempting to prove that factor spaces being connected implies the product space being connected; in particular when trying to prove the contrapositive. The proposition to be ...
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### Zariski topology is always strictly finer than product topology

Hi know there are similar question but i haven't found an answer to this particular one. Given an infinite Field $k$, show that for any n,m strictly positive integers the zariski topology on $k^{m+n}$ ...
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### Is product of perfect spaces perfect?

Let $X,Y$ be perfect topological spaces (e.g. every closed set is $G_\delta$). Is $X\times Y$ perfect? I know that this isn't true for uncountable products, for example $[0,1]$ is perfect because ...
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### Finding the basis for Product Topology

$X = \{a,b,c,d\}$ is given the topology $T_X = \{ \emptyset, \{a\}, \{a,c,d\}, \{c,d\}, X \}$ and $Y = \{1,2,3\}$ given the topology $T_Y = \{ \emptyset, \{1\}, \{1,3\}, Y \}$. Will a basis for the ...
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### Baire space is homeomorphic to countably many copies of itself

On wikipedia I found that the Baire space $\mathcal{N}$ is homeomorphic to the product of a countable number of copies of itself, however, I haven't been able to find a proof. The Baire space is ...
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### Closure of product equals product of closures: Application

One may prove that the Axiom of Choice is equivalent to the following statement $P$: If $\{(X_i,\tau_i)\mid i\in I\}$ is a system of topological sets, and $\prod_{i\in I}X_i$ is equipped with the ...
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### Show that if $A_{\alpha}$ is closed in $X_{α}$, then $\prod A_\alpha$ is closed in $\prod X_\alpha$

Show that if $A_{\alpha}$ is closed in $X_{α}$, then $\prod A_\alpha$ is closed in $\prod X_\alpha$. Please could you help me? I have no idea how to do it. I appreciate any help, hint or solution. ...
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