Intereting Posts

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$\ker \phi = (a_1, …, a_n)$ for a ring homomorphism $\phi: R \to R$
Proof of $M$ Noetherian if and only if all submodules are finitely generated
Matrix representation of the adjoint of an operator, the same as the complex conjugate of the transpose of that operator?
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To compute $\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2|^2 dz$ where $\mathcal{C}$ is the unit circle in $\mathbb{C}$
Proof of $f = g \in L^1_{loc}$ if $f$ and $g$ act equally on $C_c^\infty$
Examples of Infinite Simple Groups
Are there Hausdorff spaces which are not locally compact and in which all infinite compact sets have nonempty interior?
Hilbert class field of $\mathbb Q(\sqrt{-39})$
numbers' pattern
Is $\int_{M_{n}(\mathbb{R})} e^{-A^{2}}d\mu$ a convergent integral?(2)

Which of the following sets are compact:

- $\{(x,y,z)\in \Bbb R^3:x^2+y^2+z^2=1\}$ in the Euclidean topology.
- $\{(z_1,z_2,z_3)\in \Bbb C^3:{z_1}^2+{z_2}^2+{z_3}^2=1\}$ in the Euclidean topology.
- $\prod_{n=1}^\infty A_n$ with the product topology where $A_n=\{0,1\}$ has discrete topology.
- $\{z\in \Bbb C:|\operatorname{Re} z |\leq a \}$ for some fixed positive real number $a$ in the Euclidean topology.

$1$ is closed and bounded and hence compact,$2$ is closed but not bounded and hence not compact.

$3$ is compact by Tychonoff Theorem and $4$ is not bounded and hence not compact.

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- Question about quotient of a compact Hausdorff space
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- A theorem due to Gelfand and Kolmogorov
- How to prove a topologic space $X$ induced by a metric is compact if and only if it's sequentially compact?

Are these correct?

- In which topological spaces is every singleton set a zero set?
- Is there exist a homemoorphism between either pair of $(0,1),(0,1],$
- Looking for Cover's hubris-busting ${\mathbb R}^{N\gg3}$ counterexamples
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- If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
- Are there more general spaces than Euclidean spaces to have the Heine–Borel property?
- For $n$ even, antipodal map of $S^n$ is homotopic to reflection and has degree $-1$?
- Any idea about N-topological spaces?

That’s correct. However, 1, 2 and 4 need a proof.

All three sets are closed, being inverse images of a closed set under a continuous function.

The set in 1 is bounded, because it is contained in $[-1,1]^3$.

The sets in 2 and 4 are not bounded, because they contain element with arbitrarily large norm; can you show them?

Set 4:

Easy: you can take $z=a+bi$ with arbitrary $b$.

Set 2:

Consider $z_3=1$. Then you can take $z_2=iz_1$, for arbitrary $z_1$.

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