Intereting Posts

book recommendation for real analysis
Limit of the difference quotient of $f(x) = \frac{2}{x^2}$, as $x\rightarrow x_0$
A characterization of invertible fractional ideals of an integral domain
Classify sphere bundles over a sphere
Events in the tail $\sigma$-algebra
Upper bound on differences of consecutive zeta zeros
Symmetric Stable Distribution
Proof that $3 \mid \left( a^2+b^2 \right)$ iff $3 \mid \gcd \left( a,b\right)$
“Normalize” values to sum 1 but keeping their weights
Metrizable compactifications
For $n \in \mathbb{N}$, show that $4^n + 10 \times 9^{2n-2}$ is divisible by 7
$\sqrt{2}$ cannot represent a rational number
Proving $\kappa^{\lambda} = |\{X: X \subseteq \kappa, |X|=\lambda\}|$
Ideal of the pullback of a closed subscheme
Expected maximum absolute value of $n$ iid standard Gaussians?

Let X be a compact metrizable space. Would you help me to prove that X has a countable basis. Thanks.

- Can two different topological spaces cover each other?
- A question on star countable space
- Why $I = $ is a $1$-manifold and $I^2$ not?
- Perfect set without rationals
- Why does “separable” imply the “countable chain condition”?
- Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?
- Arbitrary intersection of closed, connected subsets of a compact space connected?
- What's wrong with this definition of continuity?
- Identifying the two-hole torus with an octagon
- Universal property of initial topology

HINT: For each positive integer $n$ let $\mathscr{U}_n=\left\{B\left(x,\frac1n\right):x\in X\right\}$; this is an open cover of $X$, so it has a finite subcover $\mathscr{B}_n$. Consider $\mathscr{B}=\bigcup_{n\in\Bbb Z^+}\mathscr{B}_n$.

- Eigenvalues of symmetric matrices are real without (!) complex numbers
- global dimension of rings and projective (flat) dimension of modules
- Prove $e^x, e^{2x}…, e^{nx}$ is linear independent on the vector space of $\mathbb{R} \to \mathbb{R}$
- Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.
- Does a regular function on an affine variety lie in the coordinate ring?(Lemma 2.1, Joe Harris)
- If $M$ and $N$ are graded modules, what is the graded structure on $\operatorname{Hom}(M,N)$?
- What consistent rules can we use to compute sums like 1 + 2 + 3 + …?
- Integer partition with fixed number of summands but without order
- How big is the size of all infinities?
- Model existence theorem in set theory
- Suppose that $(s_n)$ converges to s. Prove that $(s_n^2)$ converges to $s^2$
- Characterization of $(0 ,1)$-matrices under swap of columns or rows
- Why is it called a quadratic if it only contains $x^2$, not $x^4$?
- how to get $dx\; dy=r\;dr\;d\theta$
- Normal operator matrix norm