Intereting Posts

Extensions and contractions of prime ideals under integral extensions
Tarski Monster group with prime $3$ or $5$
Upper bound number of distinct prime factors
A certain problem concerning a Hilbert class field
Why differentiability implies continuity, but continuity does not imply differentiability?
The strong Markov property with an uncountable index set
Multiplicative group $(\mathbb R^*, ×)$ is group but $(\mathbb R, ×)$ is not group, why?
Proof of Unique Solution for Modular Exponentiation In Cryptography
How to solve following logarithmic equation: $n(n-1)3^{n} = 91854$
Intuitive use of logarithms
Classifying complex conics up to isomorphism as quotient rings of $\mathbb{C}$
Set theoretic construction of the natural numbers
Is this a good way to explicate Skolem's Paradox?
Finding a specific basis for an endomorphism
What are the properties of the roots of the incomplete/finite exponential series?

Assume we are given a space $A$ with a metric $d$. Assume $A = A_1 \times A_2 \times A_3 \cdots$, ie. $A$ is a Cartesian product of spaces $A_i$, where $i \in I$. $I$ is countable or countably infinite. Assume also that we know that each $A_i$ is homeomorphic to a space $B_i$.

Is the Cartesian product $A = \prod A_i$ homeomorphic to Cartesian product $B = \prod B_i$?

Because each $A_i$ is homeomorphic to each $B_i$, let the homeomorphism be $h_i: A_i \rightarrow B_i.$ Can I apply these in some way to an element $x \in A$ to get to $y \in B$? Like “apply $h_i$ to the $i$th element of $x$”? I dread to use projections as a projection isn’t a homeomorphism.

- In a finite abelian group of order n, must there be an element of order n?
- Prove that any finite group $G$ of even order contains an element of order $2.$
- An example of prime ideal $P$ in an integral domain such that $\bigcap_{n=1}^{\infty}P^n$ is not prime
- Galois group of $x^4-2x^2-2$
- Does $g = x^m - 1 \mid x^{mk} - 1$ for any $k \in \mathbb{N}$?
- Galois Group of $(x^2-p_1)\cdots(x^2-p_n)$

- If there are injective homomorphisms between two groups in both directions, are they isomorphic?
- Properties of the element $2 \otimes_{R} x - x \otimes_{R} 2$
- Given $n\in \mathbb N$, is there a free module with a basis of size $m$, $\forall m\geq n$?
- Is it true that a flat module is torsion free over an arbitrary ring? Does the reverse implication hold for finitely generated modules?
- Proof of the Banach–Alaoglu theorem
- Prove that any polynomial in $F$ can be written in a unique manner as a product of irreducible polynomials in F.
- Proving that infinite union of simple groups is also simple group
- The $n$-disk $D^n$ quotiented by its boundary $S^{n-1}$ gives $S^n$
- Uncountable limit point of uncountable Set (Munkres Topology)
- $(X,\mathscr T)$ is compact $\iff$ every infinite subset of $X$ has a complete limit point in $X$.

You’re quite right. We need not restrict ourselves to metric spaces or countable or finite products:

Let $A_i, B_i, i \in I$ be a family of topological spaces such that $h_i: A_i \rightarrow B_i$ is a homeomorphism for every $i$. Then $h: A = \prod_{i \in I} A_i \rightarrow B = \prod_{i \in I} B_i$ defined by $h( (x_i) ) = h(x_i)_i $ is a homeomorphism as well.

The fact that $h$ is a bijection is simple set theory. Any map $f: X \rightarrow \prod_{i \in I} X_i$ is continuous iff for every $i$, $\pi_i \circ f$ is continuous between $X$ and $X_i$, where $\pi_i$ is the projection onto the $i$’th coordinate. This is the universal property for the product topology.

From the universal property $h$ is continuous as by construction $\pi_i \circ h = h_i \circ \pi_i$, which is continuous, as a composition of continuous functions. The product of the inverses of the $h_i$ is the required continuous inverse, using the universal property for the $\prod_i A_i$ instead.

- Constructing Idempotent Generator of Idempotent Ideal
- Empty intersection and empty union
- Prove that $H$ is a abelian subgroup of odd order
- Intermediate Text in Combinatorics?
- Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$
- Show that $f^{(n)}(0)=0$ for $n=0,1,2, \dots$
- What is the Kolmogorov Extension Theorem good for?
- Subspace of $C^3$ that spanned by a set over C and over R
- How to solve this multiple summation?
- Characterize the groups $G$ for which the map $\iota: G \to G$, sending $x \mapsto x^{-1}$ for all $x \in G$, is an automorphism of $G$
- Question on showing a bijection between $\pi_1(X,x_0)$ and $$ when X is path connected.
- Can I use the “Secretary Problem” to find the worst candidate, too?
- short exact sequence of holomorphic vector bundles splits but not holomorphically, only $C^{\infty}$
- If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
- Normal distribution tail probability inequality