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As a counterexample, consider any topological space $X$ which is not metacompact. We recall that a topological space $X$ is metacompact each each open cover of $X$ has a point-finite refinement. By …

This problem has been answered by Waraszkiewicz in 1934 who proved that no metric continuum is suruniversal for all planar continua. This result was later developed by Bellamy, Krasinkiewicz, Minc, In …

After thinking some time on this question I found a relatively simple solution based on the well-known fact that all arcs in the plane are ambiently homeomorphic. Using this fact and assuming that an …

Let $C$ be a connected Tychonoff space and $a,b\in \beta C\setminus C$ be two distinct points. Let $X=\beta C$, $Y=X/\{a,b\}$ be the quotient space and $f:X\to Y$ be the quotient map. It is clear that …

Oh, sorry! I wrote this question and after some thinking found a (relatively simple) answer. Consider the set $\mathcal P$ of pairs $(A,I)$ where $A$ is a non-empty closed subset of the compact metri …

For the subspace $C_k(\omega_1;{\mathbb Z})$ of $C_k(\omega_1)$ consisting of integer-valued functions, the proof of the Lindelöf property is relatively simple. Given any open cover ${\mathcal U}$ of …

Here is (a bit lengthy and technical) proof of the Lindelof property of the function space $C_k(\omega_1)$. At first some notations. For any function $f\in C_k(\omega_1)$ and a countable ordinal $\al …

As a counterexample to this question we can consider the Katetov extension $\kappa\omega$ of the discrete space of all finite ordinals $\omega$. By definition, $\kappa\omega$ is the space of all ul …

As a counterexample to this question we can consider the Katetov extension $\kappa\omega$ of the discrete space of all finite ordinals $\omega$. By definition, $\kappa\omega$ is the space of all ul …

The second question has negative answer: just consider the unit interval $[0,1]$ and let $\mathcal S$ be the family of all closed subsets with a unique non-isolated point in $[0,1]$. The family $\math …

Since the union of two nowhere dense sets is nowhere dense, the number $\nu(X)$ is always infinite. For meager spaces it is countable. But it cannot be equal to 3.

A counterexample to this problem (with $X$ Baire and $Y$ compact) was recently constructed by Mykhaylyuk and Pol in this preprint.

Yes, there exists such a relation on $\mathbb Q$. Just use the fact that the rational projective space $\mathbb QP^\infty$ from (the answer to) this question is a countable, Hausdorff, connected (and …

This question seems to be posed too quickly (without substantial preliminary thinking) and has an (almost trivial) affirmative answer: We should prove that $\mathrm{Cont}(\mathbb R,\mathbb R)$ interse …

For polynomials with non-negative integer coefficients and no constant term, the following simple (but not obvious) fact was observed by Paulina Szczuka. Theorem. Each polynomial $f:\mathbb N\to\math …