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Can we distill the idea of "connectivity" away from their topological context and study abstract properties of "connectivity"?

I define a connective space to be a set $X$ together with a collection $\gamma$ of subsets of $X$, which we define as "connected". $\gamma$ contains every singleton subset of $X$, and for all $A, B \in \gamma$ such that $A \cap B \neq \emptyset$ we have $A \cup B \in \gamma$.

It might be interesting to study functions between connective spaces that preserve connected sets. Or, more suggestively, perhaps functions such that every pre-image of a connected set is connected... Does this exist in literature?

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The answer is yes. See this article (J. Muscat & D. Buhagiar - Connective Spaces).

Added: I did some googling and found also this preprint (S. Dugowson - On Connectivity Spaces) which examines related ideas. The references here suggest that such spaces have been studied already by R. Börger in 1983.

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    That's wonderful, thanks so much for the deep reference!2012-07-13
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    @HerngYi: I added another reference which might be useful to you.2012-07-13
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    I've seen that preprint before, but it was too dense for me to digest. My sense, though, is that it focuses on category-theoretical properties and structural similarities to links from knot theory.2012-07-13
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    I have not fully digested the Muscat & Buhagiar article, but the flavor of the article, especially towards the end, is experimental and exploratory. The bibliography of the article seems to include only texts on Analysis or Topology; do those references discuss Connective Spaces too? Or is this investigation still in its infancy, and the Muscat & Buhagiar article is one of the "definitive" expositions?2012-07-13
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    @HerngYi: Actually, I think it's currently the only exposition of this particular type of spaces, since they claim them to be new. The spaces introduced by Muscat & Buhagiar seem to be a special case (satisfying some additional properties) of the spaces studied by Börger already in 1983. So the idea of studying such spaces is at least that old.2012-07-13
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    I've digested most of the article, but doesn't their definition of the connective structure rooted at a point *exclude* the empty set, which they include in all connective structures? Admittedly, using Axiom (i) in Definition 1.1 to imply the inclusion of the empty set is rather shaky since it depends on the definition of the intersection of an empty family of sets.2012-07-14
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    I'm beginning to doubt the originality of the Muscat and Buhagiar article... the concept of touching sets mirrors the ["closeness" property](http://en.wikipedia.org/wiki/Closeness_(topology)) quite well.2012-08-10
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In my book "Topology and Groupoids" I give one definition of a connected space as follows. Let $\mathbf 2$ be the discrete topological space on the set $\{0,1\}$. Then a space $X$ is connected in the usual sense if and only if every continuous function $X \to \mathbf 2$ is constant. This definition can be used for some neat proofs: see my answer to question 90746. Part of the point is to shift emphasis from the internal properties of the space to the view of the space in the category $\mathsf{ Top}$ of spaces and continuous functions.

Notice also that a singleton space $S$ is a terminal object in $\mathsf{ Top}$ and $\mathbf 2= S \sqcup S$ (disjoint union) so one is near to a categorical definition of connectivity.