How much of general topology can be developed by taking the notion of "connected set" as the sole topological primitive Let X be an infinite regular topological space which is connected and locally connected.
Question. If no point of X is a cut point, does X always have base of connected open sets whose complements (with respect to X) are also connected?
The answer is "YES" if X is a real Banach space whose dimension is at least 2. The motivation for this question is to obtain a simple characterization of as large a class of spaces as possible, which have the property that their topology is uniquely determined when the collection of their connected subsets is specified. Note that closed connected sets can be defined in terms of connected sets. They are those connected sets whose union with the singleton of any point not belonging to them is not connected. Then we can take the complements of the closed connected sets to be the base of our topology.
 A: The answer is no. We can construct a connected and locally connected space $X$ in the plane such that:

*

*$X$ has no cut points; and

*there is a point $p\in X$ such that $X\setminus U$ is disconnected for every sufficiently small open subset $U$ of $X$ containing $p$.

Let $p=\langle 0,0\rangle$ and let $\{S_n:n<\omega\}$ be a collection of simple closed curves in the plane such that $S_n\cap S_m=\{p\}$ for all $n\neq m$, each $S_n$ intersects the circle of radius $1$ around $p$, and each $S_n\setminus \{p\}$ is open in $S:=\bigcup \{S_n:n<\omega\}$. For each $n<\omega$ connect $S_{n}$ to $S_{n+1}$ with an arc $A_n$ which has one endpoint in $S_{n}$ and the other endpoint in $S_{n+1}$, such that $A_n$ is contained in the disc of radius $1/n$ around $p$. Let $X=S\cup \bigcup \{A_n:n<\omega\}$. If $U$ is any open set containing $p$ of diameter less than $1$ then $X\setminus U$ is not connected, because for  sufficiently large $n$ the curve $S_n$ reaches outside of $U$ but its connecting arcs $A_{n-1}$ and $A_n$ are contained in $U$.
The example is Polish but non-compact.  I'm not sure if there is a compact counterexample.
