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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 …

A good starting point for studying countinua in the sense of the definition (2), i.e., compact Hausdorff spaces, is the survey paper "Continuum Theory (General)" by Ed Tymchatyn in Encyclopedia of Gen …

The answer to this question is negative and can be obtained with the help of weak P-points in compact spaces with countable cellularity. A non-isolated point $p$ of a topological space $X$ is a weak $ …

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 finally found an answer to my own question: by an old (nontrivial) result of Putcha and Weissglass, a semigroup $X$ is $E$-separated if and only if it is viable. A semigroup $X$ is viable if for any …

As a counterexample one can consider the projective space $P\mathbb R^\omega$ of the countable product of lines. This is a quotient space of the Polish space $\mathbb R^\omega_\circ=\mathbb R^\omega\s …

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 …

This is in Exercise 23.11 in the textbook of Kechris (and follows from the $\mathbf \Pi^0_3$-completeness of this space). The topological (infinite-dimensional) structure of this space is described in …

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 …