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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 …

Let $Z=\{0\}\cup\{\pm\frac1n\}_{n\in\mathbb N}$ be the sequence that converges to zero from both sides. Consider the contractible continuum $$A=(Z\times[-1,1]\times\{0\})\cup([-1,1]\times\{0\}\times\{ …

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 …

I am looking for a characterization of topological spaces $X,Y$ for which the function space $C_k(X,Y)$ is Baire. Here $C_k(X,Y)$ is the space of continuous functions from $X$ to $Y$, endowed with th …

Let $X$ be a topological space. Let $[X]^{<\omega}$ be the space of non-empty finite subsets of $X$, endowed with the Vietoris topology. For a natural number $n$ the subspace $$[X]^{\le n}:=\{A\in[X]^ …

A topological space $X$ is called $\bullet$ sequential if for each non-closed subset $A\subset X$ there exists a sequence $\{a_n\}_{n\in\omega}\subset A$ that converges to a point $a\notin A$; $\bu …

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 …

Problem. Does every compact countable space contain a non-trivial convergent sequence? This question concerns non-Hausdorff compact spaces. An example of such space is any infinite set $X$ endowed wi …

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 …

It is well-known that each uncountable compact metrizable space $X$ contains a homeomorphic copy of the Cantor cube $\{0,1\}^\omega$. What about copies of Cantor cubes of larger weight? Problem. Does …

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 …