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.
677 questions
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Product Topology alternative definition
I would like to know if the following characterisation of the product topology is correct. Would it be correct to think of the product topology as
$$
\{ \bigcup_{U \in \mathscr{E}, V \in \mathscr{F}}U ...
- 73
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74
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Extension of continuous functions from dense subsets of $\mathbb{R}^T$
Let $T$ be an uncountable index set and consider the product space $\mathbb{R}^T$ with the product topology. Suppose $Y \subset \mathbb{R}^T$ is a dense linear subspace.
Is it true that every ...
- 658
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Retrieve Manifold from Product with Circle [duplicate]
My question is as follows. Let $X, Y$ be (closed) topological manifolds (Hausdorff, with second countable basis). Suppose that $X \times S^1$ and $Y \times S^1$ are homeomorphic. Does this imply that $...
- 101
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Convergence of sequence $(1, 0, 0, \ldots)$, $(0, 1, 0, \ldots)$, $(0, 0, 1, 0, \ldots)$ in $\mathbb{R}^\omega$
So, using the definition that a sequence converges to a point if for every neighborhood of the point there exists only finitely many members of the sequence outside that neighborhood, I believe I can ...
- 1,725
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194
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Is the composition of two continuous functions a continuous operation?
I have three topological space, $\Omega$, $R$, and $S$. Then consider function spaces $\bigotimes_{\omega \in \Omega} R$ (here $\bigotimes$ is a generalized Cartesian product; not my prefered $\LaTeX$ ...
- 1,553
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1
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68
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Subspace topology on a subset of a Product space
I have a question regarding the following proposition:
Let $ X_i $ be topological spaces and $ S_i \subseteq X_i $ subspaces for $ i = 1, \ldots, n $. Also, consider the subspace $ \prod_{j=1}^n S_j \...
- 11
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2
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167
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Basis for topology on product spaces
Proposition:
Let $ \mathcal{B}_i $ be a basis for the topology on $ X_i $. Then the collection
$$
\mathcal{B} = \{ B_1 \times \ldots \times B_n \mid B_i \in \mathcal{B}_i \}
$$
is a basis for the ...
- 11
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194
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An example of an admissible topology on compacta that doesn't coincide with the compact-open topology
Given two topological spaces $X$ and $Y$, the set $Y^X=\{f:X\rightarrow Y : f \textrm{ is a function} \}$ is the set of all (not necesarily continuous) functions from $X$ to $Y$. The subsets $U^K=\{f\...
- 525
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33
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Continuity of Left but not right multiplication in Ellis Semigroup
I'm trying to read this paper by Ellis and Nerurkar:
https://www.jstor.org/stable/2001067?seq=2
So given a locally compact topological group $G$ acting on the right on a compact Hausdorff topological ...
- 547
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When is the set of continuous functions $\mathcal C(X,Y)$ equal to the set of continuous funcrtions on compacta $\mathcal K(X,Y)$?
I´m reading a chapter about topologies induced in subset of the set of all functions between two topological spaces (T.Husain, Topology and maps). Here the author introduces the sets
$$
\mathcal C(X,Y)...
- 525
5
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161
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Sequence of measurable functions from $[0,1]$ to $[0,1]$ without measurable accumulation point?
Consider the topological space $X=[0,1]^{[0,1]}$ of functions from $[0,1]$ to itself with the product topology. It is compact by Tychonoff's theorem. Consider a sequence of measurable functions $f_n\...
- 662
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3
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1k
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Torus as a product topology
I'm taking a first course in topology and we were told that the $2$-torus is the product of two circles
$$T^2 = S^1 \times S^1$$
Now I would of course expect that $T^2$ lives within $\mathbb{R}^3$, ...
- 289
1
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1
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92
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Proof that $R_0$ , $ T_1$ , and $ T_2$ Separation Axioms are Product-Stable Using Filters
I am working through an exercise on proving that the separation axioms $R_0$ , $T_1$ , and $T_2$ are product-stable, i.e., if a family of spaces $(\underline{X}_i)_{i \in I}$ satisfies one of ...
- 773
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89
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Intersection of open sets in product topology
Given a topological spaces $X$ and $Y$, I need to show that the finite intersection of open sets in the product topology $X \times Y$ is open.
Here's what I have so far. Let $u_1, \cdots , u_n$ be a ...
3
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3
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Every vector topology on $X\times Y$ is the product of the inherited topologies?
Let $X,Y$ be real (or complex) vector spaces. Let $T$ be a vector topology on $X\times Y$.
The spaces $X$ and $Y$ inherit vector topologies $T_X$ and $T_Y$ from $T$ when we identify $X$ with $X\times\{...
- 765