Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to $\mathbb{N}$. Also, the product topology, according to my textbook is as follows: Let $(X_1, \mathscr{T}_...

Let $X_1$, $X_2$, and $X_3$ be spaces. (a) Prove that $(X_1 \times X_2) \times X_3$ is homeomorphic to $(X_1 \times X_2) \times X_3$ is homeomorphic to $X_1 \times (X_2 \times X_3)$ So, I think I have the idea behind this one, but was hoping for some second opinions, critiques, or fixes. proo...

Let $X_1, X_2, Y$ be topological spaces and let $X_1 \times X_2$ be the topological space obtained by furnishing the Cartesian product set with the product topology. Let $f: X_1 \times X_2 \to Y$ be a given map. Then f is continuous iff for each $U \in Open(Y)$ and for each $(x_1, x_2) \in X_1 ...

Suppose that the metric space $(X_i,d_i)$ is topologically equivalent to $(Y_i,d'_i)$ for $i=1,2, \cdots , n$. Show that the product metric spaces $X = \prod_{i=1}^nX_i$ and $Y= \prod_{i=1}^nY_i$ are topologically equivalent. I know that since $(X_i,d_i)$ is topologically equivalent to $(Y_i,d'...

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 space $Z$ together with continuous mappings $f_i:Z\to X_i$ let me construct a categorical product ...

We know that a product of two (or finitely many) compact topological spaces is compact. And we also know that in a metric space, compactness is equivalent to sequential compactness. So a product of two sequentially compact metric spaces is sequentially compact. My question is this: Let $(X, d_...

Suppose you have a cartesian product of spaces $\prod_{\alpha\in\mathcal{A}}X_{\alpha}$ in the product topology. Choose any $\alpha\in\mathcal{A}$ . Is the following a homeomorphism of a subspace of $\prod_{\alpha\in\mathcal{A}}X_{\alpha}$ with $X_{\alpha}$? For each point $p_{\alpha}\in X_...

I am stuck on a problem about homeomorphic topological spaces and can't go on... So the problem is: If we have that $X_{1} \times X_{2}\simeq Y_{1} \times Y_{2}$ (the product of topological spaces X1 and X2 is homeomorphic to the product of Y1 and Y2), to prove is that the components might not b...

In Wikipedia and PlanetMath product of uniform spaces is defined as the weakest uniformity on the Cartesian product making all the projection maps uniformly continuous. But Springer's encyclopedia has a different (supposedly equivalent) definition. The sad part is that I don't understand their n...

Let $X,Y,Z$ be topological spaces. It is well-known that if $F:X\times Y\to Z$ is a continuous map, we can define a map $$\overline{F}:X\to C(Y,Z) \\\overline{F}(x)(y)=F(x,y)$$ where $C(Y,Z)$ is the topological space of all continuous maps from $Y$ to $Z$ equipped with the compact-open topology, ...

(If the title is unclear, I'm looking at infinite cartesian product of $\mathbb{R}$ indexed by $\mathbb{R}$.) I thought that I had reasoned this rather well, as follows: $\mathbb{R}^\mathbb{R} = \{f\mid f:\mathbb{R}\rightarrow\mathbb{R}\}$. Note that this includes functions whose range is not $...

The word products is used not only for products of numbers, functions, matrices and so on (i.e., product as a binary operation) but also for products of various algebraic structures, spaces, products in category theory etc. If I understand correctly the tag-excerpt (created by Davide Giraudo, thi...

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 space $Z$ together with continuous mappings $f_i:Z\to X_i$ let me construct a categorical product ...

I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesn't possess uncountable product of non-one point spaces. Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where $X\in\mathrm{Ob(Met)}$, $\pi_\nu\in \mathrm{Hom_{Met}}(X,X_\nu)$ is called a product of family ...

Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to $\mathbb{N}$. Also, the product topology, according to my textbook is as follows: Let $(X_1, \mathscr{T}_...

(If the title is unclear, I'm looking at infinite cartesian product of $\mathbb{R}$ indexed by $\mathbb{R}$.) I thought that I had reasoned this rather well, as follows: $\mathbb{R}^\mathbb{R} = \{f\mid f:\mathbb{R}\rightarrow\mathbb{R}\}$. Note that this includes functions whose range is not $...

There seems to be considerable support for taking on this tag (+8/-0), which is "high", for questions about tagging). This answer intends to poll the specific approach I suggested. Concretely, I suggest to: Introduce congruences-and-remainders as a synonym for modular-arithmetic; Create a con...