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.

Filter by
Sorted by
Tagged with
1 vote
2 answers
57 views

Let $A\subset\Bbb R^m$ and $B\subset\Bbb R^n$ be closed subsets. Then $A \times B$ is a closed subset of $\Bbb R^{m+n}$

I am preparing for my exam right now and therefore practicing by doing some exercises. I need help for the following task: Let $A\subset\Bbb R^m$ and $B\subset\Bbb R^n$ be closed subsets. Then $A \...
user avatar
0 votes
1 answer
32 views

The minimality of product topology.

Let $(X,\mathcal O_X), (Y, \mathcal O_Y)$ be two topological spaces, and let $\mathcal B=\{ U\times V \mid U\in \mathcal O_X, V\in \mathcal O_Y \}$. Product topology $\mathcal O$ is a family of sets ...
user avatar
  • 2,221
-3 votes
0 answers
21 views

is open set in subspace equal to intersection open set in whole space with same subspace? [closed]

Is open set in subspace equal to intersection open set in whole space with same subspace?
user avatar
1 vote
1 answer
65 views

A topology product problem with sequence of continuous and unbounded functions

Consider $X=\{f:\mathbb{Q}\rightarrow \mathbb{R}\}$ (set of any functions), with the product topology. Is true that for any $f\in X$, there are $(f_n)_{n\in \mathbb{N}}$ sequence of continuous and ...
user avatar
0 votes
1 answer
37 views

CW structure on the product $X\times Y$ of two CW-complexes $X$ and $Y$

I do understand that the $k$ -cells of $X\times Y$ are $\{e^i_X\times e^{k-i}_Y\}$ where $e^i_X$ is an $i$-cell of $X$ and $e^{k-i}_Y$ is a $k-i$ cell of $Y$ . I have the attaching maps from the CW-...
user avatar
  • 708
14 votes
0 answers
204 views

Can ($X^I$, product topology) and ($X^I$, box topology) be homeomorphic for some nontrivial $X$ and infinite $I$?

Let $X$ be a nontrivial topological space, $I$ be a infinite set, we can endow $X^I$ (the set of all functions $I\to X$) with either the product topology or the box topology. We know that the box ...
user avatar
3 votes
1 answer
77 views

Continuity of function wrt intersection of family of topologies

Let $X$ be a set and $\{\tau_i\}_{i\in I}$ a family of topologies on $X$. If the set map $f:X\times X\to X$ is continuous w.r.t. $\tau_i$ ($X\times X$ given product topology) for all $i$,then will $f$ ...
user avatar
  • 447
0 votes
2 answers
43 views

differentials of smoothly varying family of maps

Let $F:N\times M \to M'$ be a smooth map, which we interpret as a "smooth" family of maps $M \to M'$, parametrized by N, so we have a map $F(y,\cdot):M \to M' \: \forall y \in N$. Show that ...
user avatar
  • 341
0 votes
1 answer
47 views

Doubt in homeomorphism of product spaces [closed]

Let $X,Y,Z$ topological spaces. Suppose that product spaces $X \times Y$ and $X \times Z$ are homeomorphic. It's true that $Y$ and $Z$ are homeomorphics? Since that the projection map is only ...
user avatar
6 votes
1 answer
61 views

Vector space translation continuous implies addition continuous?

In here, it is proved if a topology on a vector space makes the addition function continuous, then the translation is also continuous everywhere. My question is whether the inverse is still true: Let $...
user avatar
  • 61
0 votes
1 answer
44 views

Continuity of a function from $R^\omega \to R^\omega$ for uniform, product and box topology.

Bear with me, because neither my English nor my Topology is as good as it should be. So I was studying topology and found an online list of problems, I was trying to do this one: Let $f: R^\omega \to ...
user avatar
0 votes
1 answer
54 views

An uncountable product of $\mathbb{R}$ with itself is not metrizable (product topology).

An uncountable product of $\mathbb{R}$ with itself isn't metrizable in the product topology I wrote what I believe is an incorrect proof of this statement since I feel like I ignored the fact that ...
user avatar
  • 57
0 votes
1 answer
36 views

Is every function from $R^\omega$ to $R^\omega$ with uniform topology continuous?

So I was studying topology and I came across the next theorem: A function $f: X \to Y$ is continous iff for every $\epsilon>0$ there is $\delta>0$ such that if $d_x(x,y)<\delta$ then $d_y(f(x)...
user avatar
0 votes
1 answer
25 views

Criterion for continuity of a function on the Cartesian product

Let $X$ and $A$ be topological spaces. Endow $X\times X$ with the product topology. Let $$f\colon X\times X\to A$$ be a (not necessarily continuous) map. For each $x\in X$, define functions $f_{1,x},...
user avatar
  • 4,150
2 votes
1 answer
29 views

The subspace topology on $Y$ will be if $\mathbb{R}^2$ has the product topology $\mathbb{R}_u \times \mathbb{R}_u$

This is related to an earlier problem I submitted on this site. Let $Y \subset \mathbb{R}^2$ be the x-axis. I'm trying to figure out what the subspace topology on $Y$ will be if $\mathbb{R}^2$ has the ...
user avatar
3 votes
1 answer
74 views

Are Borel measurable functions closed in pointwise topology?

Let $X$ be a metrizable space. The Lebesgue-Hausdorff theorem states that the minimal class $\mathcal{C}$ of functions $f :X \to \mathbb{R}$ closed under pointwise limits of sequences, such that $C(X) ...
user avatar
  • 1,779
3 votes
3 answers
94 views

Constructing the topological product space from categorical universal property

Let $X,Y$ be topological spaces. I want to construct the categorical product, i.e. the product space, $X\times Y$ using the universal property of products. To be clear, I don't want to construct the ...
user avatar
  • 5,713
3 votes
1 answer
151 views

Can a Torus be a submanifold of a Sphere?

If I describe a $2$-torus in $4D$, as the product of two independent circles $S^1\times S^1$. Can the resulting torus live on the $4D$-embedded sphere $S^3$? I want to confirm points of my torus can ...
user avatar
3 votes
0 answers
76 views

Prove that X is a Hausdorff space if and only if the diagonal is a closed set in its product topology

So I was trying to do this,and asked my professor to tell me if it was correct, but he wouldn't look at my work, here is my attemp to a solution. Proof: Let $\Delta$ be the diagonal. $$\implies:$$ Let ...
user avatar
1 vote
1 answer
73 views

Are these spaces zero-dimensional? What are some characterizations of zero-dimensional spaces?

Disclaimer: By a zero-dimensional space I mean a topological space having a base of sets that are at the same time open and closed in it. At my university, we discussed zero-dimensionality and ...
user avatar
1 vote
1 answer
141 views

The infinite product of connected spaces is connected in the product topology.

Is my proof correct? Any general feedback on proof writing style is also much appreciated! 1. Theorem The infinite product of connected spaces is connected in the product topology. In particular, let $...
user avatar
  • 119
0 votes
1 answer
58 views

$\left\{0\right\}\times (0,1) \not \in \mathbb R^2$

So I was taking my topology class and professor said it's trivial to see that $\left\{0\right\}\times (0,1)$ is not a member of the usual topology $ \mathbb R^2$, is there a way to prove this or is it ...
user avatar
2 votes
1 answer
120 views

Find closure of sets w.r.t product topology?

Consider the cartesian product $$X= ⨉_{r \in[0,1]} \ [0,\infty) = \{\text{all functions } f:[0,1] \rightarrow [0,\infty)\}$$ with the (product) topology generated by the projections $$ S = \{p_r^{-1}(...
user avatar
  • 691
0 votes
2 answers
34 views

How do I show this statement about convergence in filters?

I have the following problem: Let $\{(M_i,\tau_i)\}_{i\in I}$ be nonempty topological spaces where $I$ is arbitrary but non empty. Let $M=\prod_{i\in I} M_i$. Let $F$ be a filter on $M$ and denote by ...
user avatar
  • 1,029
0 votes
1 answer
39 views

How do I prove this statement about the continuity of the product map?

I have the following problem: Leg $I$ be a nonempty index set and let $(M_i,\mathfrak{M_i})$ and $(N_1,\mathfrak{N_i})$ be topological spaces. Moreover let $f_i:M_i\rightarrow N_i$ be maps. Finally ...
user avatar
  • 1,029
0 votes
1 answer
57 views

Density of sets in the product space $X= \Bbb R^{\Bbb R}$

Let $X= \Bbb R^{\Bbb R}$ and suppose that $X$ has the product topology. Let $A$ be the set of functions $f: \Bbb R \to \Bbb R$ for which $f(x) \in \Bbb Q$ for all $x \in \Bbb R$ and $f(x)=0$ for all ...
user avatar
  • 81
0 votes
1 answer
22 views

Is restriction of a projection to a connected subset a quotient map?

Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}_Z$ be the product topology on $Z$, $f : Z \to X$ be defined by $f(x, y) = x$, and $C \subset Z$ ...
user avatar
  • 1,235
4 votes
1 answer
55 views

Prove, that the topology of $X$ coinduced by maps $f_i$ from the usual topology of $\Bbb R$ is finer than the one $X$ inherited from $\Bbb R^{\Bbb N}$

Let $X$ be the union of the coordinate axes in the product set $\Bbb R ^{\Bbb N}$, that is, the set of those points $x$, for which $x_n \ne 0$ for at most one $n \in \Bbb N$. For each $i \in \Bbb N$ ...
user avatar
  • 143
0 votes
2 answers
37 views

Let $X= \Bbb R^{\Bbb N}$ and let $A\subset X$ be the set of bounded sequences. Is $A$ open/closed in the product topology of $\Bbb R^{\Bbb N}$?

Let $X= \Bbb R^{\Bbb N}$ and let $A\subset X$ be the set of bounded sequences. Is $A$ open/closed in the product topology of $\Bbb R^{\Bbb N}$? I have hard time with these questions related to the ...
user avatar
  • 143
3 votes
1 answer
64 views

Let $g,h : \Bbb R \to \Bbb R$ be any functions. Define $F: \Bbb R^{\Bbb R} \to \Bbb R^{\Bbb R}$ as $F(f)(t)=g(t)f(h(t))$. Show that $F$ is continuous.

Let $g,h : \Bbb R \to \Bbb R$ be any functions. Define $F: \Bbb R^{\Bbb R} \to \Bbb R^{\Bbb R}$ as $F(f)(t)=g(t)f(h(t))$. Show that $F$ is continuous in the product topology. Is $f$ the variable that'...
user avatar
  • 143
0 votes
1 answer
36 views

How do I prove that a filter converges in a cartesian product?

I have the following problem: Let $\{(M_i, \mathfrak{M}_i)\}$ be nonempty topopological spaces. And let $M=\prod_{i\in I} M_i$. Let $\mathfrak{F}$ be a filter on $M$ and so $\mathfrak{F}_i=(\pi_i)_* \...
user avatar
  • 1,029
0 votes
2 answers
34 views

Path connected product topology implies every topology is path connected

Suppose that $(X_i, \tau_{X_i})$ are path-connected topological spaces for all $i \in I$. I know that the product $\Pi_{i \in I}X_i$ with its product topology is path-connected. But is the converse ...
user avatar
  • 996
0 votes
1 answer
30 views

Confusion over the metric in the product of two topological groups

$\newcommand{\m}{\,\operatorname{mod}}$I cite this text, page $38$ - although they reference results from earlier on in the text: Some definitions: A topological dynamical system $(K;\phi)$ is a non-...
user avatar
  • 8,444
-1 votes
1 answer
42 views

Continuity of a bilinear form with respect to weak$^*$ topology [closed]

Let $X$ be a normed linear space and let $X^*$ be its topological dual. The bilinear form $\psi:X\times X^*\to F$ is defined by $$\psi(x,x^*)=x^*(x).$$ Is $\psi$ continuous with respect to the ...
user avatar
  • 1,228
1 vote
0 answers
63 views

One question in Spanier's Algebraic Topology Theorem 1.4.12

In Spanier's Algebraic Topology Chapter 1 Section 4, he says that $Z_f \times I$ has the topology coinduced by the two maps as the following, which I do not understand: Why $Z_f \times I$ equipped ...
user avatar
  • 139
0 votes
1 answer
45 views

Coinduced topology and product topology

Let $f_1:X_1\rightarrow X$ and $f_2:X_2\rightarrow X$ be two maps from two topological spaces to a set X. We equip $X$ with the topology coinduced by $f_1$ and $f_2$, that is to say, a subset $U\...
user avatar
  • 139
0 votes
1 answer
72 views

Why isn't the product $\sigma$-algebra defined as the pre-image $\sigma$-algebra of the canonical projections

Let $f:X_1\times X_2\rightarrow Y$ be a mapping and let $\mathscr{B}$ be the $\sigma$-algebra on $Y$. Now, we know that the pre-image is a $\sigma$-algebra, so in this case $f^{-1}(\mathscr{B})$ must ...
user avatar
1 vote
1 answer
74 views

Stone–Čech compactification of a Tychonoff space can be taken as closure of diagonal mapping image in Tychonoff cube [closed]

Let $X$ - Tychonoff topological space. Show that Stone–Čech compactification of $X$ can be obtained by taking the closure of the image of the space $X$ under the mapping $\Delta_{f\in C(X, I)}f$ in ...
user avatar
  • 51
0 votes
1 answer
25 views

Confusion regarding the proof steps of the topology generated by a subbasis

Munkres, page 88, Theorem 15.2 The author outlines the proof (given above) that the subbasis $S$ generates the topology $\tau'$ which is same as the product topology $\tau$. The proof can be broken ...
user avatar
  • 101
0 votes
0 answers
51 views

Integrability of the mean sum (or sample mean)

Suppose that $X \in \mathcal{X}$ and $X\sim P$, is integrable, i.e., $\int |X| \mathrm{d}P<\infty$. Let $(X_1,\ldots,X_n)$ be a vector of $n$ i.i.d. instances of $X$. Hence, $X^n\sim P^{\otimes n}$....
user avatar
  • 333
0 votes
1 answer
41 views

Munkres topology Exercise $2.22$ Question $1$

The latter picture is what I currently have for this one, But I have no faith that it is correct, I am having a hard time with composing functions. Need an answer to learn from.
user avatar
1 vote
0 answers
36 views

When does $f(x_i,y) \to f(x,y)$ follow from $f(x_i,y_i) \to f(x,y)$ and $y_i \to y$?

Say I have topological spaces $X$, $Y$ and $Z$, and a continuous map $f : X \times Y \to Z$. Say also that I have sequences $(x_i)$ in $X$ and $(y_i)$ in $Y$, along with points $x \in X$ and $y \in Y$ ...
user avatar
0 votes
0 answers
46 views

Limit points of a product space

I'm new to topology and am trying some exercises in Gaal's Point Set Topology (p 63). How do I show that the set of limit points of the product space $(A × B)' = (A' × \overline B\ ) ∪ ( \overline A\ ×...
user avatar
0 votes
1 answer
71 views

Show that the Euclidean topology in a product of spaces Vector matches the product of Euclidean topologies.

A topological isomorphism between two topological vector spaces is an application between the two that is both isomorphism and homeomorphism. If $V$ is a vector $\mathbb K$-space of finite dimension $...
user avatar
0 votes
0 answers
56 views

what does "pasting" and "cut" mean in function (topology)

There is an example that explain this, but I still have 3 question: Let $f: [0, 1] ∪] 2, 3] → [0, 2]$ the application given by $f (x) = x$ if $0 ≤ x ≤ 1$ $f(x)=x - 1$ if $2 <x ≤3$ ...
user avatar
1 vote
1 answer
40 views

Why the set of models of a proposition is a closed set in the product topology

I'm trying to understand the topological proof of compactness of propositional logic, so here's the part (I think) I understand: We let $\mathcal A$ be the set of propositional variables, $\Gamma$ ...
user avatar
  • 4,300
0 votes
1 answer
58 views

How do I show that the filter is convergent in a cartesian product?

I have the following problem: Let $\{(M_i,T_i)\}_{i\in I}$ be non-empty top. spaces where $I$ is an arbitrary non empty index set. Moreover let $F$ be a filter on $M=\prod_{i\in I}M_i$ and denote $...
user avatar
  • 1,284
0 votes
1 answer
63 views

Induced topology on Hilbert cube

I am given an exercise which I have some trouble understanding: Let $X$ be the Hilbert cube, i.e. the product space $X = [0,1]^{\mathbb{N}}$ and let it be equipped with the product topology, thus $$B ...
user avatar
  • 634
2 votes
1 answer
39 views

Convergence in distribution on product space (definition) [closed]

Let $S$ and $T$ be seperable metric spaces, $(X_n)_{n\in\mathbb{N}}$ a $S$-valued sequence of random variables and $(Y_n)_{n\in\mathbb{N}}$ a $T$-valued sequence of random variables Is convergence in ...
user avatar
  • 143
0 votes
1 answer
41 views

Essential Surface in $F\setminus\!\{\text{point}\} \times S^1$

So I am reading about essential surfaces, and I know that there is an essential torus in a punctured surface times $S^1$. I just don't see it? The only torus I can think of would be one $\partial$-...
user avatar
  • 323

1
2 3 4 5
11