# Tagged Questions

For questions about the structure of product space, in the context of topology or measure theory for example. Use other tags to indicate the context.

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### Urysohn Metrization Theorem contradiction (uniform topology homeomorphic to product topology)?

The theorem states that if $F$ is regular and has a countable basis, then it is metrizable. In Munkres' proof of this theorem, he gives a function (homeomorphism) $F:X \rightarrow [0,1]^\omega$ that ...
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### Lindelöf space with compact space [closed]

This is one of the exercises from Munkres'. Show that if $X$ is Lindelöf and $Y$ is compact, then $X\times Y$ is Lindelöf.
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### Fubini's theorem for a measure on product space, which is not a product of measures

Let $X,Y$ be some nice measurable spaces (I'm interested in $[0,1]$ so we can assume compact, etc.). Let $\mu$ be a measure on $X\times Y$ (again, assume it's a nice probability measure, or even ...
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### Question about continuity in the box topology

I have two question regarding the following example in Munkres (1)Why "if $f^{-1}(B)$ were open it would contain some interval $(-\delta,\delta)$ about 0. (2)My second question is somewhat broad, but ...
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### Box topology definition

Munkres defines the box topology as shown below. I am trying to understand how come an element of basis as defined is even subset of $\prod_{\alpha \in J}X_{\alpha}$. If we get an element of the basis,...
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### Example Infinite Product of Compact Metric Spaces that is not itself compact [closed]

Can anyone think of an infinite product space of compact metric spaces that us not itself compact?
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### Second derivative/tangential map on a product manifold

Short version: Given $f: M_1 \times M_2 \longrightarrow N$ what is the second tangential $TTf$ expressed in partial tangentials on $M_1, M_2$? The details: Let $M_1, M_2, N$ be manifolds. The partial ...
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### How do I prove that the product topology is the unique topology for which projections are well-behaved?

This is a question from a practice qualifying exam. Let $X, Y, T$ be topological spaces. Define $p_X: X\times Y\to X$ and $p_Y:X\times Y\to Y$ to be the projection maps. Then $f:T\to X\times Y$ is ...
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### General set in product space approximated by rectangle sets

Let $(E^k,\mathcal{E}^k,\mu^k)$ be a product measure space. By a rectangle set in $E^k$, we mean a set of the form $A_1\times\ldots\times A_k$ where each $A_i\in \mathcal{E}$. My question is, for ...
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### Prove that d is a continuous real valued function on $M \times M$ [duplicate]

Let $(M,d)$ be a metric space. Let $M \times M$ be the product space, where $d$ is defined (earlier in the book as ) $d((x_1,x_2),(y_1,y_2))=d_1(x_1,y_1)+d_2(x_2,y_2)$ where $d_1$ and $d_2$ are ...
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### Products containing the same digits

I am looking for $d_1$ and $d_2$, where $d_1$ is a two digit integer and $d_2$ is a three digit integer, such that $d_1\cdot d_2$ is a product that contains the exact same digits as $d_1$ and $d_2$. ...
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### How can I prove that $(X,τ)$ is a Hausdorff topological space?

Let $(X_1,τ_1)$ is a Hausdorff topological space and $(X_2,τ_2)$ is a Hausdorff topological space and $X=X_1\times X_2$ and $τ$ The product topology How can I prove that $(X,τ)$ is a Hausdorff ...
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### Probability of Union of Events in a Probability Product Space By Counting Event Size

This question is about probability of union of events in a probability product space. Let’s say a fair die is thrown twice and we’re interested to find out probability of getting face value one in 1st ...
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### If $(X,\tau)$ is Hausdorff, is $X\times X$?

If $(X,\tau)$ is Hausdorff, is the product topology $X\times X$? I have a feeling that some part of $\Bbb R$ to be Hausdorff, means there is isolation for all elements on the line, and if that is the ...
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### Compact subsets of the space of real functions $\mathbb{R}^\mathbb{R}$

I was suprised that this question hasn't been asked - or maybe it was, but asked differently. Anyway, I want to characterize the compact sets in the space of real functions $\mathbb{R}^\mathbb{R}$ (...
Let $J$ be a given (countably or uncountably infinite) index set. Let $\{\ X_\alpha \ \colon \ \alpha \in J \ \}$ and $\{\ Y_\alpha \ \colon \ \alpha \in J \ \}$ be collections of topological spaces....