6:02 AM
I will be retagging the questions tagged as products+general-topology with .
At the moment there are 12 questions.
1

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 $(X_1, \mathscr{T}_... ^ I am not sure about this one. It seems more about cardinality rather than general topology. (In which case is a correct tag.) 1 Let$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)$So, I think I have the idea behind this one, but was hoping for some second opinions, critiques, or fixes. proo... 1 Let$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 ...

0

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 $(Y_i,d'... 0 Let$\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$ii) Find a base for the product topology on$X\times Y$I found the product topology as$\tau^1=\{X\times Y,...

2

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 ...

1

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 $(X, d_... 2 Suppose you have a cartesian product of spaces$\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_...

1

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...

0

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...

5

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, ...

And again there is one which is more about inrinite product of sets than about topological products. (Although the product of metric spaces is mentioned in the post.) So this one probably should not be retagged either:
3

(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 $... I will also repeat the link to the relevant discussion on meta. Although so far it does not seem as any kind of consensus, I think that at least the question which fit into already existing tag can be retagged. 4 The word products is used not only for products of numbers, functions, matrices and so on (i.e., product as a binary operation) but also for products of various algebraic structures, spaces, products in category theory etc. If I understand correctly the tag-excerpt (created by Davide Giraudo, thi... 6:47 AM This question is related to categorical products. So if we introduce a separate tag of that (and it will not be a synonym of ), it should be retagged again: 2 Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces$X_i$and any topological space$Z$together with continuous mappings$f_i:Z\to X_i$let me construct a categorical product ... The same applies to this question (it is about categorical products): 8 I need to prove that category$\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)$is called a product of family ... 7:20 AM @MartinSleziak I'm in favor of having a seperate tag for gcd/lcm next to . @barto Thanks for the reply. I suppose you have already upvoted quid's post, which suggests exactly this. Indeed, I have. We will see how this will pan out. Note that I suggested a tag synonym but that was only for the sake of consistency after reading this discussion at Tag Management 2015. When they were made synonyms, the decision was based on a post which now has score +8: (11/-3): meta.math.stackexchange.com/questions/10998/on-the-gcd-tag/… Yes, I know about the synonyms. All of them have score +2 at the moment" math.stackexchange.com/tags/divisibility/synonyms 8:03 AM So now there are only questions tagged products+general-topology: 1 Prove that if$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}_...

3

(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 $... To me it seems that they are more suitable for tag, if we decide to create such tag as a result of the discussion on meta. 3 hours later… 10:38 AM For a brief period we had tag, but it was removed. I wonder whether tag for limits and colimits in the sense of category theory would be suitable. Some users seem to tag such questions with limits+category-theory, which is IMO not very good. 11:05 AM I don't think it would be harmful to create, say, . But perhaps we want limits and colimits in 1 tag? Honestly I don't know what these things are so I'm not sure if it makes sense. @barto I would definitely prefer the same tag for both of them. Of course, the explanation that it includes colimits might be in tag-info. (It is not necessary to include it into the tag name.) Or we could even create a tag for colimits and make it a synonym. (If we go ahead and create such tag.) On meta.MO they have a tag called colimits which is clearly about colimits in the sense of category theory. The tag limits on that site seems to contain mix of variuos things. I would guess that there are quite a lot of question on this topic on MSE. If I search for limit among the questions tagged category-theory: math.stackexchange.com/search?q=limit+%5Bcategory-theory%5D 1 hour later… 12:23 PM 0 Via a question I took a look at the polynomial-tag and found that the tag wiki says this: Polynomials are expressions like$15x^3 - 14x^2 + 8\$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for ro...

1 hour later…
1:29 PM
@MartinSleziak When retagging gcd-related questions, consider adding when appropriate.

1:50 PM
Thanks for the reminder. I will try not to forge about it.

1 hour later…
2:51 PM
I couldn't stand anymore so I have suggested to make it a synonym of .

Making the two tags synonymous was basically what LF suggested in the original thread. (Although he suggested it together with the change of the name of the tag):
4

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...

1 hour later…
4:19 PM
I know, but the discussion has been going on for so long I thought it was time to take it to a next step.