Intereting Posts

$2^a +1$ is not divisible by $2^b-1$.
Suppose $\lim \limits_{n \to ∞} a_n=L$. Prove that $\lim\limits_{n \to ∞} \frac{a_1+a_2+\cdots+a_n}{n}=L$
Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)
How to find the original coordinates of a point inside an irregular rectangle?
Is the fundamental group of every subset of $\mathbb{R}^2$ torsion free?
Fourier transform as a Gelfand transform
Is it possible to find radical solution of $\sin(5\beta)+\sin(\beta)=1$ and finding radical approximation of $\pi$
What is a good asymptotic for $f_n = f_{n-1}+\ln(f_{n-1})$?
How to prove that $1+2=3, 4+5+6=7+8,… $ ad infinitum?
History of “Show that $44\dots 88 \dots 9$ is a perfect square”
Why does having fewer open sets make more sets compact?
Find the maximum and minimum radii vectors of section of the surface $(x^2+y^2+z^2)^2=a^2x^2+b^2y^2+c^2z^2$ by the plane $lx+my+nz=0$
Proof that $\sum\limits_{j,k=1}^N\frac{a_ja_k}{j+k}\ge0$
How find limit $\displaystyle \lim_{n\to\infty}n\left(1-\tfrac{\ln n}{n}\right)^n$
Another quadratic Diophantine equation: How do I proceed?

I have a question about product topology.

Suppose $I=[0,1]$, i.e. a closed interval with usual topology. We can construct a product space $X=I^I$, i.e. uncountable Cartesian product of closed interval. Is $X$ first countable?

I have read Counterexamples of Topology, on item 105, it is dealing with $I^I$. I do not quite understand the proof given on the book. Can someone give a more detail proof?

- $g$ a restriction of a homeomorphic function $f$, $g$ also homeomorphic?
- Quasicomponents and components in compact Hausdorff space
- Every neighborhood of identity in a topological group contains the product of a symmetric neighborhood of identity.
- Prove that Euclidean, spherical, and hyperbolic 3-manifolds are THE only three-dimensional geometries that are both homogeneous and isotropic.
- Lower Semicontinuity Concepts
- Raising a partial function to the power of an ordinal

- How far is being star compact from being countably compact？
- An open subset $U\subseteq R^n$ is the countable union of increasing compact sets.
- Is the Euler characteristic $\chi =2$ for the prism with a hole?
- closed bounded subset in metric space not compact
- $|f(x)-f(y)|\le(x-y)^2$ without gaplessness
- Prove that the identity map $(C,d_1) \rightarrow (C,d_\infty)$ is not continuous
- Is the boundary of a connected set connected?
- Prove that $(X\times Y)\setminus (A\times B)$ is connected
- How to show 2 bases generate the same topology?
- Differentiable Manifold Hausdorff, second countable

It is not.

Let’s look at open sets containing 0 (sequence of 0s).

We will argue by contradicton, so suppose there is a local neighbourhood basis $U_i$ for 0. Every such $U_i$ contains some $V_i = \prod_{r} V_{i,r}$ where $V_{i,r} = I$ for almost every $r$, $V_{i,r}$ open. (By definition of product topology).

Let’s look at the set of all $r$ that $V_{i,r} \neq I$ for some $i$.

This set is countable, because it’s a countable sum of countable sets. So it’s not the whole of $I$. Let’s choose some $r_0$ outside this set.

Let $H = \prod_r H_r$ where $H_{r_0} = [0, 1/2)$ and $H_{r} = I$ otherwise.

Then $H$ is an open set containing $0$, not contained in any of $U_i$, contradicting first-countability at 0.

Let $x \in I^I$ and assume $I^I$ has a countable basis $\langle A_n \rangle_{n\in \omega}$ at $x.$

Let $s_n$ be the set $i\in I$ such that the $i$-th coordinate projection map $\pi_i: A_n \rightarrow I$ is not surjective. As each $A_n$ is open, the set $s_n$ is finite for all $n.$ It follows $s := \cup_{n\in \omega} s_n$ is countable.

As $I$ is uncountable, it follows $I\setminus s$ is nonempty. Choose an element $i\in I\setminus s.$ Define for each $j\in I$ a set $X_j$ such that $X_j = I$ if $j \neq i$ and $(\pi_i(x) – 1/2,\pi_i(x) + 1/2) \cap I$ otherwise.

Then $X := \prod_{j \in I} X_j$ is open and contains $x$ but is not contained in $A_n$ for any $n.$ This contradicts the fact $\langle A_n \rangle_{n\in \omega}$ was assumed to be a basis at $x.$

It follows $I^I$ is not only not first countable but $I^I$ doesn’t have a countable basis at any point.

- Should $\mathbb{N}$ contain $0$?
- Why does this least-squares approach to proving the Sherman–Morrison Formula work?
- How to get rid of the integral in this equation $\int\limits_{x_0}^{x}{\sqrt{1+\left(\dfrac{d}{dx}f(x)\right)^2}dx}$?
- Show that if $fd'=f'd $ and the pairs $f, d $ and $f',d' $ are coprime, then $f=f' $ and $d=d' $.
- How do you calculate how many decimal places there are before the repeating digits, given a fraction that expands to a repeating decimal?
- To find the nilpotent elements of $\Bbb Z_n$ and also the number of nilpotent elements of $\Bbb Z_n$.
- Degree of continuous maps from S1 to S1 – Two equivalent properties
- A possible vacuous logical implication in Topology
- Modules over commutative rings
- Can a ring without a unit element have a subring with a unit element?
- Limit of $\sqrt{4x^2 + 3x} – 2x$ as $x \to \infty$
- What precisely is a vacuous truth?
- Question on the normal closure of a field extension
- How to find area under sines without calculus?
- Moriarty's calculator: some bizarre and deceptive graphical anomalies