Intereting Posts

how to calculate the integral of $\sin^2(x)/x^2$
Induction for statements with more than one variable.
No surjective bounded linear map from $\ell^2(\mathbf{N})$ to $\ell^1(\mathbf{N})$
A formula for heads and tail sequences
Question about the cardinality of a space
About the second fundamental form
Prime dividing the binomial coefficients
The Cauchy-Schwarz Master Class, Problem $1.2$
$f,g,h$ are polynomials. Show that…
Quartic diophantine equation: $16r^4+112r^3+200r^2-112r+16=s^2$
Does set theory help understand machine learning or make new machine learning algorithms?
measure of open set with measure Haar
Fubini's theorem for conditional expectations
Why is $(3,x^3-x^2+2x-1)$ not principal in $\mathbb{Z}$?
Tiling of regular polygon by rhombuses

Let $X$ be a Moore space and $e(X)=\omega$. Is it metrizable?

**What I’ve tried:** I list these facts:

1 A space $X$ is a Moore space iff $X$ is a $\sigma$-space and a $p$-space.

- How to prove that compact subspaces of the Sorgenfrey line are countable?
- Example of topological spaces with continuous bijections that are not homotopy equivalent
- Complement is connected iff Connected components are Simply Connected
- Star-shaped domain whose closure is not homeomorphic to $B^n$
- Is the long line completely uniformizable?
- Constructing a choice function in a complete & separable metric space
2 If $X$ is a $p$-space, then $nw(X)=w(X)$.

3 A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network.

Since if $X$ is a $\sigma$-space and $e(X)=\omega$, we know that $nw(X)=\omega$, see the proof. So by 2, we can conclude that $w(X)=nw(X)=\omega$, and hence it is metrizable.

However I’m not sure. Thanks for any help.

- Closed subset of compact set is compact
- Is it possible to determine if you were on a Möbius strip?
- Conditions that ensure that the boundary of an open set has measure zero
- Continuous extension of a real function
- Projection map being a closed map
- Manifold Boundary versus Topological Boundary.
- Sufficient condition to inscribe a polygon inside another one
- Do simply connected open sets in $\Bbb R^2$ always have continuous boundaries?
- Proving that $S=\{\frac{1}{n}:n\in\mathbb{Z}\}\cup\{0\}$ is compact using the open cover definition
- Equicontinuity on a compact metric space turns pointwise to uniform convergence

Here are the references.

A Moore space is a *regular* developable space. [Gru, 1.3] A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network. [Gru, 4.3] Every Moore space $X$ is a $\sigma$-space. [Gru, 4.5] Moreover, a space $X$ is a Moore space iff $X$ is a $\sigma$-space and a $p$-space (or a $w\Delta$-space). [Gru, 4.7.(i)] If $X$ is a $\sigma$-space and $e(X)=\omega$, we know that $nw(X)=\omega$, see the proof. If $X$ is a $p$-space, then $nw(X)=w(X)$. [Gru, 4.2] By Nagata-Smirnov Theorem [Eng, 4.4.7], a topological space $X$ is metrizable iff $X$ is regular and has a $\sigma$-locally finite base. In particular, a regular second countable space is metrizble.

**References**

[Eng] Ryszard Engelking. *General Topology* (Russian version, 1986).

[Gru] Gary Gruenhage, *Generalized Metric Spaces*, in K. Kunen and J. E. Vaughan, Handbook of set theoretic topology, Elsevier, 1984.

- When does the set enter set theory?
- What is the mechanism of Eigenvector?
- What is “Approximation Theory”?
- $x^3+y^4=7$ has no integer solutions
- Suppose that $(s_n)$ converges to s. Prove that $(s_n^2)$ converges to $s^2$
- Simple and intuitive example for Zorns Lemma
- Multiplicative norm on $\mathbb{R}$.
- Why is $\cos(x)^2$ written as $\cos^2(x)$?
- Dini's Theorem. Uniform convergence and Bolzano Weierstrass.
- Number of horse races to determine the top three out of 25 horses
- What is the importance of the Collatz conjecture?
- Trace Class: Decomposition
- Are there books introducing to Complex Analysis for people with algebraic background?
- What is the significance of reversing the polarity of the negative eigenvalues of a symmetric matrix?
- How many solutions does this equation have $x^2 \equiv 1017 (\mod 2^k)$