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If X is a topological space and Y is any set, then we call a family $\mathcal {F}$ of maps from X to Y nowhere constant if for every non-empty open set U in X there is $f \in \mathcal {F}$ with $|f[U]|> 1$, i.e., f is not constant on U. We prove the following result that improves several earlier results in the literature.
If X is a topological space for which $C(X)$, the family of all continuous maps of X to $\mathbb {R}$, is nowhere constant and X has a $\pi $-base consisting of connected sets then X is $\mathfrak {c}$-resolvable.
Edited by
Rob Waller, NHS Lothian,Omer S. Moghraby, South London & Maudsley NHS Foundation Trust,Mark Lovell, Esk and Wear Valleys NHS Foundation Trust
Well-connected systems that safely and ethically share information have a lot of potential to improve care and safety. Due to systems having developed organically and over long periods of time, the reality today is more piecemeal. There are several current initiatives to develop and improve the situation; the main barriers are often managerial or ethical rather than technical. Increasingly, the focus is on moving data out of big silos like hospitals to places where it can be accessed (when appropriate) by others.
British power at India’s northwest and northeast frontiers was only occasionally predicated on categorising and codifying, emanating more often from indeterminacy and upheaval. These were spaces of productive difficulties for colonial administrators, who prized as well as feared the supposed unruliness of uplands and deserts at the state’s fringes. This chapter provides a theoretical outline of how India’s frontiers became spaces of scientific and governmental exception, situating the book’s core arguments in relation to scholarship on power, knowledge, territory, and borderlands. It proposes that although internally fragmented by social structures, terrain, and colonial categories, colonial India’s frontiers took shape through ideational and material connections and comparisons. Following increasingly intense interventions from the later 1860s, by the turn of the twentieth century frontiers were established as crucial spaces of imperial power, science, and self-fashioning.
We prove that a connected, countable dense homogeneous space is $n$-homogeneous for every n, and strongly 2-homogeneous provided it is locally connected. We also present an example of a connected and countable dense homogeneous space which is not strongly 2-homogeneous. This answers in the negative Problem 136 ofWatson in the Open Problems in Topology Book.
We consider an unordered graph where there is one arc emanating from each node. We suppose that the arc that emanates from i will go to j with probability Pj. The probability that the resultant graph is connected and a recursive formula for the distribution of the number of components it possesses are derived.
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