If a separable dense in itself metric space is not a union of countably many nowhere dense subsets, then its $C_p$-space is not subsequential.

For a space Z let 𝒦(Z) denote the partially ordered set of all compact subspaces of Z under set inclusion. If X is a compact space, Δ is the diagonal in X², and 𝒦(X²∖Δ) has calibre (ω₁,ω), then X is metrizable. There is a compact space X such that X²∖Δ has relative calibre (ω₁,ω) in 𝒦(X²∖Δ), but which is not metrizable. Questions of Cascales et al. (2011) concerning order constraints on 𝒦(A) for every subspace of a space X are answered.

Following Preuss' general connectedness theory in topological categories, a connectedness concept for approach spaces is introduced, which unifies topological connectedness in the setting of topological spaces, and Cantor-connectedness in the setting of metric spaces.

In this paper we complete the characterization of those $f$, $\mu$ and $\nu$ such that $${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\int }_{B}f(x,w(x))d\mu +\nu (B)$$ is $\mathrm{\Gamma }(L^2{(\mathrm{\Omega })}^{-})$ limit of a sequence of obstacles ${\parallel w\parallel }_{H^1(\mathrm{\Omega })}^2+{\mathrm{\Phi }}_h(w,B)$ where

We compare several conditions sufficient for maximal resolvability of topological spaces. We prove that a space $X$ is maximally resolvable provided that for a dense set $X_0\subset X$ and for each $x\in X_0$ the $\pi$-character of $X$ at $x$ is not greater than the dispersion character of $X$. On the other hand, we show that this implication is not reversible even in the class of card-homogeneous spaces.

We prove that every compact space $X$ is a Čech-Stone compactification of a normal subspace of cardinality at most $d{\left(X\right)}^{t\left(X\right)}$, and some facts about cardinal invariants of compact spaces.

We show that a regular totally ω-narrow paratopological group G has countable index of regularity, i.e., for every neighborhood U of the identity e of G, we can find a neighborhood V of e and a countable family of neighborhoods of e in G such that ∩W∈γ VW−1⊆ U. We prove that every regular (Hausdorff) totally !-narrow paratopological group is completely regular (functionally Hausdorff). We show that the index of regularity of a regular paratopological group is less than or equal to the weak Lindelöf...

We show that if we add any number of Cohen reals to the ground model then, in the generic extension, a locally compact scattered space has at most ${\left(2^{\aleph ₀}\right)}^V$ levels of size ω. We also give a complete ZFC characterization of the cardinal sequences of regular scattered spaces. Although the classes of regular and of 0-dimensional scattered spaces are different, we prove that they have the same cardinal sequences.

Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put ${}_{\lambda }\left(\alpha \right)=s\in \left(\alpha \right):s\left(0\right)=\lambda =min\left[s\right(\beta ):\beta <\alpha ]$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $\lambda ₀i>\lambda ₁>...>{\lambda }_{n-1}$ and ordinals $\alpha ₀,...,{\alpha }_{n-1}$ such that $\alpha =\alpha ₀+\cdots +{\alpha }_{n-1}$ and $f=f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i{\in }_{{\lambda }_i}\left({\alpha }_i\right)$. Under GCH we prove that if α < ω₂ then (i) ${}_{\omega }\left(\alpha \right)=s{\in }^{\alpha }\omega ,\omega ₁:s\left(0\right)=\omega$; (ii) if λ > cf(λ) = ω, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega ₁-closedin\alpha$; (iii) if cf(λ) = ω₁, ${}_{\lambda }\left(\alpha \right)=s{\in }^{\alpha }\lambda ,\lambda ⁺:s\left(0\right)=\lambda ,s^{-1}\lambda is\omega -closedandsuccessor-closedin\alpha$; (iv) if cf(λ) > ω₁, ${}_{\lambda }\left(\alpha \right){=}^{\alpha }\lambda$. This yields a complete characterization of the classes (α) for all α < ω₂,...