Search Results

It is well-known that a compact Hausdorff $X$ space is scattered if and only if admits no continuous maps onto the unit interval $[0,1]$. Surprisingly, but I cannot find a good reference to this well …

Trying to characterize categories equivalent to the category of sets, I have discovered (for myself) that instead of requiring that the coprojection morphism $\mathsf{true}:1\to \Omega=1\sqcup 1$ is a …

I need to cite the classical Zsigmondy Theorem, which was proved in 1892. Is there any modern reference to this theorem? I mean some standard textbook in Number Theory containing this theorem together …

Given a triangle $\Delta$ with sides of length $a\le b\le c$, consider the number $$q=\frac{a^4+b^4+c^4}{(a^2+b^2+c^2)^2}$$ and observe that $\frac13\le q\le\frac12$ and the extremal values of $q$ cha …

Definition. A semigroup $X$ is called $E$-separated if for any distinct idempotents $x,y\in X$ there exists a homomorphism $h:X\to Y$ to a semilattice $Y$ such that $h(x)\ne h(y)$. Observe that $X$ is …

I need a reference to the following (known?) Fact. Every topological vector space $X$ over the field of real numbers is topologically isomorphic to a linear subspace of the Tychonoff product of li …

Definition. Let us define a Banach space $X$ to be co-Sobczyk if every linear bounded operator $T:Z\to c_0$ defined on a separable subspace $Z$ of $X$ extends to a bounded operator $\bar T:X\to c_0 …

Let $B$ be a compact convex set on the complex plane, containing zero in its interior. The boundary $\partial B$ of $B$ has the polar parametrization $\mathbf p:\mathbb R\to \partial B$ assigning to e …

I am writing a paper on K-analytic spaces and need the following known characterization. Theorem. For a regular topological space $X$ the following conditions are equivalent: (1) $X$ is a continuous …

I need a good reference (desirably some textbook in Number Theory) to the following known result, attributed to Gauss in Wikipedia. Theorem (Gauss). Let $p$ be a prime number, $k\in\mathbb N$ and $\m …

I am writing a paper on the topology of the Golomb space and need a good (standard) reference to the following General Chinese Remainder Theorem. For integer numbers $a_1,\dots,a_n$ and positive in …

Working with non-regular topological semigroups, my collegue Oleg Gutik discovered a special space $H$ which we named Gutik's hedgehog. It is homeomorphic to the space $$H:=\{(0,0)\}\cup\{(\tfrac1n,0) …

I am interested in the history related to algebraic numbers and have two questions: Who first proved that algebraic numbers form a field? Who first proved that algebraic numbers form an algebraicall …

Please help me to find a proper reference to the following infinite version of the Sunflower Lemma. Lemma. Let $n\in\mathbb N$. Every infinite family of $n$-element sets contains an infinite subfa …

For a cardinal $\kappa$ let $[\kappa]^{<\kappa}$ denote the family of subsets of cardinality $<\kappa$ in $\kappa$. The family $[\kappa]^{<\kappa}$ is endowed with the partial order of inclusion. A si …