TITLE: Abstract connectedness
QUESTION [15 upvotes]: Is there an abstract structure that characterizes connectedness, analogously to how topological spaces characterize continuity?
Here's one way to make this question more precise: if $(X,T_X)$ is a topological space with underlying set $X$ and topology $T_X$, then consider the pair $(X,C_X)$ where $C_X$ is the set of connected subsets of $X$.  A continuous map $f:(X,T_X) \to (Y,T_Y)$ has the property that the direct image of a connected set is connected.  Thus, we have a functor from Top to the category whose objects are pairs $(X,C_X)$ of a set equipped with a set of subsets, and whose maps $f:(X,C_X)\to (Y,C_Y)$ are functions $f:X\to Y$ such that $f(U) \in C_Y$ for all $U\in C_X$.  Is there a naturally defined full subcategory of the latter category that is "close" to the image of Top?
There are undoubtedly other ways that one could make this question more precise; if you have a better suggestion feel free to make it.

REPLY [2 votes]: Let me answer as to how well the connected subspaces of a space determine the topology. I claim that for a large class of spaces, the collection of all connected subsets of the space determines the topology on $X$. However, the functions between nice connected spaces that map connected sets to connected sets are generally not continuous.
The following lemma states that the closure of a connected subspace of a regular space $X$ is determined by the collection of all connected subspaces of $X$. In particular, the closed connected subspaces of a regular space $X$ are determined by the collection of all the connected subspaces of $X$.
$\mathbf{Lemma}$ Suppose that $X$ is a regular space and $A\subseteq X$ is a connected subset. Then $\{a\}\cup A$ is connected if and only if $a\in\overline{A}$.
Furthermore, if $A\subseteq B\subseteq X$, then $B\subseteq\overline{A}$ if and only if whenever $A\subseteq C\subseteq B$ then $\overline{C}$ is connected.
The proof of the above lemma is straightforward.
We shall say that a point $x$ is a connected space $X$ is a nbd-cut-point if there is some open neighborhood $U$ of $x$ such that whenever $V$ is an open set with $x\in V\subseteq U$, then $X\setminus V$ is not connected. Then it is easy to show that a connected space $X$ has no nbd-cut-points if and only if every closed subspace of $X$ is the intersection of connected closed subspaces of $X$. I should also mention that I made up the notion of a nbd-cut-point in order to answer this question, but the notion of a nbd-cut-point is closely related to the notion of a cut-point since every cut point is a nbd-cut-point.
$\mathbf{Proposition}$ Suppose that $X,Y$ are connected regular spaces with no nbd-cut points. If there is a bijection $f:X\rightarrow Y$ so that if $C\subseteq X$ then $C$ is connected if and only if $f[C]$ is connected, then $X$ and $Y$ are homeomorphic.
However, I must mention that even for connected regular spaces with no nbd-cut-points, there are non-continuous maps where the image of a connected space is connected. For example, define an equivalence relation on $\mathbb{R}$ where $x\simeq y$ iff $x-y\in\mathbb{Q}$. Then there is a surjective function $f:\mathbb{R}/\simeq\rightarrow\mathbb{R}^{2}$ since $|\mathbb{R}/\simeq|=|\mathbb{R}^{2}|$. Define a mapping $g:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ by letting $g(x,y)=f([x])$ and define $h:\mathbb{R}\rightarrow\mathbb{R}^{2}$ by letting $h(x)=f([x])$. Then $h[U]=\mathbb{R}^{2}$ for each nonempty open set $U$. I claim that $g$ maps connected subsets to connected subsets. Let $\pi_{1},\pi_{2}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ be the projection maps where $\pi_{1}(x,y)=x,\pi_{2}(x,y)=y$. Suppose that $C\subseteq\mathbb{R}^{2}$ is connected. Then $\pi_{1}[C]$ is also connected, so $\pi_{1}[C]$ is either a singleton or an interval. If $\pi_{1}[C]$ is a singleton, then $g[C]$ is a singleton in $\mathbb{R}^{2}$. However, if $\pi_{1}[C]$ is an interval, then $g[C]=\mathbb{R}^{2}$. In either case, the function $g$ maps connected sets to connected sets, but the function $g$ is very far from being continuous. It therefore seems as if the notion of continuity cannot be described in terms of connectedness unless one describes the open or closed sets in terms of connectedness and directly translates the notion of continuity into a notion of a connected mapping.