# 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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### What is a finite subcover of $[0,1]^{[0,1]}$?

According to Tychonoff's theorem, under the standard topology, $[0,1]^{[0,1]}$ is compact. However, I cannot think of a finite subcover of this space. Also, how does this reconcile with the fact that ...
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### Is this proof regarding product of connected spaces correct?

Let $X,Y$ be connected spaces, and consider their product $X\times Y$. I want to show that their product is connected. The posts I've read here regarding this question often include creating "slices" ...
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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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### In general, why is the product topology not equal to the box topology

I am trying to understand a counter-example showing that the box topology and product topology are not equal. Here it is: Let $\tau$ and $\tau'$ be the product and box topologies respectively. Let ...
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### How is the absolute value on a product space defined?

I want to show that a map from the cantor set $C$ to its product $C\times C$ is homeomorphic, and I am stuck at continuity because in the epsilon delta proof I am not sure how to deal with the ...
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I am trying to prove that the product of T1 spaces is also T1. Here is a proof, is it correct? $\{ X_i \}_{i \in I}$ are T1 $\Rightarrow$ $\prod_{i \in I} X_i$ is T1 Proof: Let $\bar{x} = ( ... 1answer 58 views ### How to find subbase and base for$X\times Y$? Let$\tau :=\{X,\emptyset,\{a\},\{b,c\}\} $on$X=\{a,b,c\}$and$\tau^*:=\{Y,\emptyset,\{u\}\}$on$Y:=\{u,v\}$i) Find a subbase for the product topology on$X\times Y$ii) Find a ... 1answer 22 views ### continuity of a function f = (f_1,f_2) in a product topology if f_1 and f_2 are continous Say$X$,$Y_1$and$Y_2$are topological spaces. Let$f_1 \; X \to Y_1$and$f_2 \; X \to Y_2$. If$f\; X \to Y_1 \times Y_2 f(x) = (f_1(x), f_2(x))Y_1 \times Y_2$is a topological space with ... 1answer 50 views ### Prove$ \varphi ( x ) = \lambda ( Q_x ) $is$ \mathcal{X} $-measurable I'm reading Thoerem 8.6, page 163, Chapter 8 from "Real and Complex Analysis (Third Edition)" by Walter Rudin; it seems that Rudin never proves that$\lambda(Q_x)$(defined below) is a ... 2answers 202 views ### Continuity of sum/product using characteristic property of product topology I'm self studying Lee's Introduction to Topological Manifolds, and I'm familiarising myself to the characteristic/universal property view of topology. One of the exercises in chapter 3 goes as ... 2answers 361 views ### Proof: Categorical Product = Topological Product Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces$X_i$and any topological ... 1answer 283 views ### A generalization of the generalized tube lemma I am trying to prove the following generalization of the generalized tube lemma: Let$\{X_t\}_{t \in T}$be a family of Hausdorff spaces and$\prod_{t \in T}A_t$be a compact subset of$X=\prod_{t ...
I am new to advanced mathematics and I recently started reading a book on topology. I am struggling to understand what it is saying in this paragraph. This is what it says: Let $E_i$ ...
### Product topology on $X \times Y$ the smallest topology when $f(x, y) = x$ and $g(x, y) = y$ are continuous functions?
$X$ and $Y$ are topological spaces and let $f : X \times Y \to X$ and $g : X \times Y \to Y$ be maps such that $f(x, y) = x$ and $g(x, y) = y \ \forall (x, y) \in X \times Y$ . Show that the product ...