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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77 views

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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1answer
64 views

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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1answer
13 views

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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1answer
69 views

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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1answer
33 views

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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1answer
21 views

Cartesian product of diffeomorphic Lie groups

Let $A,B,$ and $C$ be Lie groups. If $A$ and $B$ are diffeomorphic, are the Cartesian products $A \times C$ and $B\times C$ diffeomorphic? In other words, does $A \cong B$ imply $A \times C \cong B\...
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1answer
29 views

How to show a graph is homeomorphic to a subset of a product space?

$X, Y$ are topological spaces and $X \times Y$ is given the product topology. The subspace $ G \subseteq X \times Y$ is defined as: $ G =$ { $(x,y) \in X \times Y | y = f(x)$ }. How can one show that ...
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2answers
42 views

$f: \prod_{i \geq 1} \{0,1,2,3\} \rightarrow \prod_{i \geq 1} \{0,2\}$ is a homeomorphism

We view $X = \{0,1,2,3\}$ as a topological space equipped with the discrete topology. $f: \prod_{i \geq 1} X \rightarrow \prod_{i \geq 1} \{0,2\}$ is a homeomorphism. So f maps a sequence $(a_{i})_{i \...
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0answers
58 views

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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2answers
41 views

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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1answer
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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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1answer
33 views

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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1answer
46 views

Is $f(x) = (x,x)$ not continuous in this topology?

Let $S$ be the unit circle with the usual topology and let $\mathbb{R}$ be equipped with the discrete topology, then consider $S \times \mathbb{R}$ with the product topology, and consider the map $f(x)...
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1answer
45 views

Quotient space Hausdorff/$T_2$?

Let $n$ natural number and $X = [0,1] \times \{1,...,n\}$ with the topology $\tau \times \tau_D$, where $\tau$ is the usual topology of $\mathbb{R}$ induced by $[0,1]$ and $\tau_D$ is the discrete ...
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2answers
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closure of A in the product space

For each $\alpha\in I=[0,1]$, let $X_{\alpha} = \{0,1\}$ be the discrete topology. For every $\Delta\subseteq I$, define $f_\Delta=(x_\alpha)\in\prod_{\alpha\in I}X_\alpha$, where $x_\alpha=1$ if $\...
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1answer
37 views

$(x,y)\mapsto f(x)g(y)$ is continuous

Let $X$ and $Y$ be topological spaces; $X\times Y$ carries the product topology. Can someone provide me with an easy argument that for $f\in C(X,\mathbb{R})$ and $g\in C(X,\mathbb{R})$ the mapping $$ ...
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1answer
38 views

Prove that $\overline{A \times B}=\overline{A} \times \overline{B}$

How I can prove that $\overline{A \times B}=\overline{A} \times \overline{B}$? I start proving that $\overline{A} \times \overline{B} \subseteq \overline{A \times B}$. Taking $(x,y) \in \overline{A} ...
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1answer
84 views

Proving Theorem 19.2 in <Topology> by Munkres

Theorem 19.2 in Munkres Suppose the topology on each space $X_{\alpha}$ is given by a basis $\mathcal{B}_{\alpha}$. The collection of all sets of the form $$\prod_{\alpha \in J} B_{\alpha}$$ where $B_{...
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1answer
68 views

All the product topologies

I wonder what are all the topologies on the product of spaces that satisfy certain properties. For example, let $(X_i,\mathcal{T}_i)_{i\in I}$ be a family of topological spaces and let $X:=\prod_{i\in ...
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1answer
58 views

Valid proof technique for proving compactness of a topological space?

My goal is to prove that if given that the product space $X \times Y$ of two topological spaces is compact, then $X$ and $Y$ are compact. My idea for proof technique is as follows. Prove that the ...
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1answer
37 views

Does the converse of the statement hold?

I've proved the following statement: "Given $(Z, T_Z)$ the product topological space of $(X, T_X)$ and $(Y, T_Y)$. If $A \in T_X$ and $B \in T_Y$ then $A \times B \in T_Z$". Converse ...
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0answers
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Conclusion about corollary V.6.1 in Schaeffer & Wolff (Topological Vector Spaces 2nd edition)

Currently, I am stuck with following the final conclusion in the proof of corollary V.6.1 in Schaeffer & Wolff's Topological Vector Spaces, 2nd edition, pp. 230-231. Probably it is trivial and I ...
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1answer
19 views

Inverse of projection map in product topology which defines sub-basis.

I have $(X,\tau_X)$ and $(Y,\tau_Y)$ - two topological spaces. Consider $U\subseteq X$ and $V\subseteq Y$ and two projection maps $\pi_X: X\times Y \to X$ and $\pi_Y: X \times Y \to Y$. Then it is ...
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0answers
23 views

Proof Verification of Countable Product of Separable Spaces

I know this question is frequently asked on this website, but I'm not sure if my proof is correct: Question. Prove that if each factor space $X_n$, $n=1,2,\cdots$, is separable, then so is the ...
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1answer
23 views

Approaching Compact Sets of the Euclidean Space with the Product Topology

I was working on Croom's Principles of Topology with the product topology; and came across this problem: Knowing that a subset $A$ of $\mathbb R$ is compact if and only if it is closed and bounded, ...
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0answers
24 views

Compact submanifold of $S^n \times R^m$

On exercise 6 of chapter 11 of "Comprehensive Introduction to Differential Geometry" by Spivak, one is asked to apply the exact sequence for the pair $(S^n \times R^m,\{p\}\times R^m)$ to ...
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2answers
84 views

Proof $(X×Y)-(A×B)$ is connected [Contradiction]

Let $A$ be a proper subset of $X$, and let $B$ is a proper subset of $Y$. If $X$ and $Y$ are connected, show that $$(X×Y)-(A×B)$$ is connected. I know there's already a question answering this problem,...
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0answers
20 views

Product topology and nets

I just learned about nets, but I'm confused about when to use them. More precisely when combined with products. Let $(X_i)_{i\in I}$ be a family of topological spaces and consider $X = \prod_{i \in I} ...
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1answer
51 views

Product topology for $\mathbb{R}^{\mathbb{N}}$ and generating subbasis

The product topology on $\mathbb{R}^{\mathbb{N}}$ is defined to be the topology generated by the subbasis: $$S_n := \{ \pi_n ^{-1} (U) : U \subseteq \mathbb{R} \text{ open}\} $$ where $\pi_n$ are ...
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1answer
40 views

Diagonal is closed in $(X, \tau_1) \times (X, \tau_2)$ (both are Hausdorff) [duplicate]

From Willard's General Topology: Let $\tau_1$ and $\tau_2$ be Hausdorff topologies on the same set $X$. Let $\tau_3 = \tau_1 \cap \tau_2$. If $(X, \tau_3)$ is Hausdorff, then the diagonal is closed ...
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1answer
66 views

Why is an element in the countable product of real numbers regarded as a sequence?

A sequence of real numbers is a just an enumerated list of real numbers. Now in my lecture notes it is mentioned that an element of the countable product topological space $\mathbb{R}^w$ (Tychonoff ...
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0answers
45 views

What are all the topological conjugacies between the same adding machines?

Let $(\Sigma_k, f)$ be an adding machine on $k$ symbols. That is, $\Sigma_k$ is the set of all sequences of numbers $0, 1, \ldots, k - 1$ and $f: x = (x_0, x_1, \ldots) \mapsto (x_0, x_1, x_2, \ldots) ...
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1answer
42 views

Showing a subset is open with respect to a topology

I have asked and read similar questions, but I am still somewhat confused on the notion of "a set being open with respect to some topology". My task is concerning the box topology and the ...
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1answer
32 views

Showing a set is open with respect to box vs product topology

I am currently studying topology with Munkres and I am asking for some general proof techniques and clarification rather than just posting my question and wait for a solution. I am given that the real ...
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1answer
46 views

Searching a contraexample

Is finite complement topology on $\mathbb{R^2}$ the sameas the product topology on $\mathbb{R}$ with the finite complement topology.If $\mathbb{R_{fc}}$ denote the finite complement toplogy in $\...
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2answers
83 views

Product topology and continuity

Hi guys I'm new to topology and was asked to prove the following, of which I am having troubles with: Let $F:X \times I \rightarrow Y$ be a continous function. For each $t \in I$ let $f_{t} = F(x,t)$. ...
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0answers
34 views

Function of products and product of functions are equal?

Say $h:\prod X_{\alpha}\rightarrow \prod X_{\alpha}$ and $\space h(x)=(h_{\alpha}(x_{\alpha}))$, $U = \prod U_{\alpha}$ is an open set in $\prod X_{\alpha}$. It's $h(U) = \prod (h_{\alpha}(U_{\alpha}))...
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1answer
60 views

Showing $\mathbb{R}^2 \cong \mathbb{R}\times (-\frac{\pi}{2}, \frac{\pi}{2})$ using $\tan^{-1}$

just a quick, if not obvious, question here -- since the $\arctan$ function from $\mathbb{R}$ to $(-\pi /2, \pi /2)$ is a homeomorphism, does it naturally follow that the function $f: \mathbb{R}^2 \to ...
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2answers
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An infinite product of locally metrizable spaces is locally metrizable(in the product topology)?

Locally metrizable means that each point has a neighborhood that is metrizable in the subspace topology. I think it can be proved that a finite product of locally metrizable spaces is locally ...
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1answer
53 views

The product of limit point compact spaces is limit point compact?

Assuming the Tychonoff theorem that arbitrary product of compact spaces is compact, is it true that the product of limit point compact spaces is limit point compact? If not, are there any ...
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1answer
28 views

On Product topology and standard topology on Real space

The standard topoplogy $O_{\text {stantard } \mathbb{R}^{d}}$ on $\mathbb{R}^{d}$ is defined as the collection of sets $$U \in O_{\text {stantard } \mathbb{R}^{d}}$$ such that $$\forall p \in U: \...
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1answer
33 views

Understanding measurability on a product space

I am having trouble understanding the concept of borel algebra on $\mathbb R^n$ and how it applies to measurable functions. I learned in class that the sigma algebra on $\mathbb R^n$ is generated by $...
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1answer
70 views

Product space definition

Herebelow, all passages in Billingsley $(1995)$ to get to definition of product spaces: The standard construction of the general process involves product spaces. Let $T$ be an arbitrary index set and ...
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2answers
98 views

why $ \bigcup (X \cup Y)= X\times Y $?

I have some confusion in Munkres topology . My confusion is given below marked in red colour My attempt : Here ${\bigcup}_{x\in X} T_x = \bigcup(X \times b) \cup (x \times Y)= \bigcup (X \cup Y)$ ...
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1answer
44 views

Prove on metrization of uncountable product [duplicate]

I am given the following problem: Show that an uncountable product of unit intervals is not first countable, and thus not metrizable. My answer would be that a), since the elements of the neighborhood ...
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2answers
40 views

Basic topology: proof verification of a set not being open in the product topology

Let $A=\displaystyle\prod_{i\in[0,1]}\left(0,1\right)$ and $X=\displaystyle\prod_{i\in[0,1]}\mathbb{R}$. Is $A$ open in $X$ with respect to the product topology on $X$? Here is what I did: By ...
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0answers
56 views

Relation between universal properties and topological invariants.

I don't really have any experience working with universal maps or adjoint functors (well, neither with categories, really. Only some basic definitions and intuitions). However, I have the feeling that ...
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1answer
56 views

Prove that the product of compact sets in a product space is contained in a basic open set.

The full question: Given topological spaces $X$ and $Y$, and compact sets $A \subseteq X$ and $B \subseteq Y$, and open set $W \subset X \times Y$ such that $A \times B \subseteq W$, then there exists ...
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1answer
79 views

Prove that $\mathbb{R}\not\cong X\times X$ for any $X$ [duplicate]

Prove that $\not\exists$ a topological space $X$ such that $\mathbb{R}$ is homeomorphic to $X\times X$. My approach was to prove by contradiction: say $f:\mathbb{R}\rightarrow X\times X$ be such a ...
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1answer
44 views

Proving Tychonoff Theorem using the Wallace Theorem

Prove the Tychonoff theorem using the Wallace theorem and Kuratowki theorem Wallace Theorem: If $A_\alpha$ is a compact subset of $X_\alpha$ for all $\alpha \in A$, then, for every open subset $W$ of ...

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