# Comparing Different Notions of Unicoherence in the Plane

Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech cohomology and homology. There is a related notion of 'open unicoherence' but there has been little study devoted specifically to it, and my question concerns the difference between the two concepts.

A topological space $X$ is unicoherent if for every pair of closed, connected subsets $A, B$ such that $A \cup B = X$, we have $A \cap B$ is connected. We say that a space is open unicoherent if the same is true but with 'closed, connected subsets' replaced by 'open, connected subsets.' In fact, the differences between open unicoherence and unicoherence are highly related to the comparison between shape theory and Cech homology, but my question is more down to Earth.

From now on assume that $X$ is a compact, connected subset of the plane, i.e. a planar continuum. In the following paper, examples are given of planar continua which are unicoherent, but not open unicoherent, and vice-versa: https://biblat.unam.mx/en/revista/anales-del-instituto-de-matematicas-unam/articulo/a-survey-on-unicoherence-and-related-properties

We can define a stronger property as well. Say that a planar continuum is hereditarily unicoherent if each of its subcontinua (including itself) is unicoherent. Define hereditary open unicoherence similarly. Then, is it known whether either of these properties implies the other for planar continua?

Question 1) If $X$ is a hereditarily unicoherent planar continuum, is $X$ (hereditarily) open unicoherent?

Question 2) If $X$ is a hereditarily open unicoherent planar continuum, is $X$ (hereditarily) unicoherent?

In trying to think of a counterexample for (2), we are looking for a continuum whose connectivity properties among its subcontinua is not well-represented by the connectivity properties of its connected open subsets. In an indecomposable continuum all connected, open subsets are dense, so they seem like a good class of candidates. But the examples I tried, like the pseudo-arc and buckethandle continuum, don't supply any counterexamples as far as I can tell. All arc-like continua are hereditarily unicoherent and planar, but it seems they might also be hereditarily open unicoherent. All dendrites (in fact all dendroids) are known to satisfy both properties, e.g. see Nadler's Continuum Theory, ch. 10 and its exercises. Note that any non-trivial hereditarily unicoherent continuum, open or otherwise, is one-dimensional, lest it contain a circle.

Question 3) Are all arc-like continua open unicoherent?

This is surely known, but I have trouble thinking of how to prove it. I am not very familiar with characterizations of connected open subsets of arc-like continua. Since subcontinua of arc-like continua are arc-like, this is equivalent to asking if they are hereditarily open unicoherent. If this is false then it gives a negative answer to #1.

There are also the following similar questions. They are, a priori, slight weakenings of the first two, though they seem liable to be equivalent.

Question 4) If $X$ is unicoherent and hereditarily open unicoherent, is it hereditarily unicoherent?

Question 5) If $X$ is open unicoherent and hereditarily unicoherent, is it hereditarily open unicoherent?

However, I don't see an easy way to show that these are false if the first two are.