Connectedness and compactness on standard sets

We present a nonstandard characterization of connected compact sets. (C) 2010 WILEY-VCH Verlag GmbH & Co KGaA. WeinheimCEOCFCTFEDER/POCI 201

Connectedness modulo an ideal

For a topological space $X$ and an ideal $\mathscr{H}$ of subsets of $X$ we introduce the notion of connectedness modulo $\mathscr{H}$. This notion of connectedness naturally generalizes the notion of connectedness in its usual sense. In the case when $X$ is completely regular, we introduce a subspace $\gamma_{\mathscr H} X$ of the Stone--\v{C}ech compactification $\beta X$ of $X$, such that connectedness modulo ${\mathscr H}$ is equivalent to connectedness of $\beta X\setminus\gamma_{\mathscr H} X$. In particular, we prove that when ${\mathscr H}$ is the ideal generated by the collection of all open subspaces of $X$ with pseudocompact closure, then $X$ is connected modulo ${\mathscr H}$ if and only if $\mathrm{cl}_{\beta X}(\beta X\setminus\upsilon X)$ is connected, and when $X$ is normal and ${\mathscr H}$ is the ideal generated by the collection of all closed realcompact subspaces of $X$, then $X$ is connected modulo ${\mathscr H}$ if and only if $\mathrm{cl}_{\beta X}(\upsilon X\setminus X)$ is connected. Here $\upsilon X$ is the Hewitt realcompactification of $X$.Comment: 28 page

Connectedness modulo a topological property

Let ${\mathscr P}$ be a topological property. We say that a space $X$ is ${\mathscr P}$-connected if there exists no pair $C$ and $D$ of disjoint cozero-sets of $X$ with non-${\mathscr P}$ closure such that the remainder $X\backslash(C\cup D)$ is contained in a cozero-set of $X$ with ${\mathscr P}$ closure. If ${\mathscr P}$ is taken to be "being empty" then ${\mathscr P}$-connectedness coincides with connectedness in its usual sense. We characterize completely regular ${\mathscr P}$-connected spaces, with ${\mathscr P}$ subject to some mild requirements. Then, we study conditions under which unions of ${\mathscr P}$-connected subspaces of a space are ${\mathscr P}$-connected. Also, we study classes of mappings which preserve ${\mathscr P}$-connectedness. We conclude with a detailed study of the special case in which ${\mathscr P}$ is pseudocompactness. In particular, when ${\mathscr P}$ is pseudocompactness, we prove that a completely regular space $X$ is ${\mathscr P}$-connected if and only if $cl_{\beta X}(\beta X\backslash\upsilon X)$ is connected, and that ${\mathscr P}$-connectedness is preserved under perfect open continuous surjections. We leave some problems open.Comment: 12 page

The connected Vietoris powerlocale

The connected Vietoris powerlocale is defined as a strong monad Vc on the category of locales. VcX is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are “strongly connected”. A product map ×:VcX×VcY→Vc(X×Y) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then VcX is overt. For the localic completion of a generalized metric space Y, the points of are certain Cauchy filters of formal balls for the finite power set with respect to a Vietoris metric. \ud Application to the point-free real line gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem. \ud \ud The work is topos-valid (assuming natural numbers object). Vc is a geometric constructio