End points of continua

Question

Whyburn (1942) defined an end point $x$ of a continuum $X$ to be any point having arbitrarily small neighborhoods each of whose boundaries contains a single point. Thus, he defines an end point locally.

Other definitions exist, however. I have seen an end point defined as any point $x \in X$ with the property that, if two subcontinua $A, B \subseteq X$ contain $x$, then one of these subcontinua must be a subset of the other one. This seems to be a non-local definition of end point.

Under Whyburn's definition, a triod has three end points, but under the second definition, it has none.

Am I missing something here? What is currently the most widely used definition of end point of a continuum? I shall appreciate a comment or clarification.

Answer 1

I´m not an expert in continua theory, but I would say that whatever definition you choose for end point, a triod should have three of them (as it happens with Whyburn´s definition).

The second definition that you mention appears (perhaps for the first time?) in R.H. Bing´s "Snake-like continua", where he shows:

For a point $p$ in a Snake-like continuum $M$, the following are equivalent:

1. If each of two subcontinua of $M$ contains $p$, one of the subcontinua contains the other.
2. For each positive number $\epsilon$ there is an $\epsilon$-chain covering $M$ such that only the first link of the chain contains $p$.

An $\epsilon$-chain is a finite sequence of open sets of diameter less than $\epsilon$ such that only consecutive terms (links) can have non-empty intersection. A continuum is Snake-like (or chainable) if it can be covered by an $\epsilon$-chain for each positive $\epsilon$.

Property 2 is Bing´s definition of end point and it seems to me that it might coincide with Whyburn´s definition. So I would say that property 1 should be used as a definition of end point only for chainable continua. However it has been used also for general continua (see for instance "Continua with a dense set of end points" by Charatonik and Mackowiak).

Finally, you might be interested in this early paper: "Concerning end points of continuous curves and other continua" by H.M. Gehman, where the (non-)equivalence of various definitions is studied.

Answer 2

There are indeed quite a number of definitions of end-points, or of terminal points (the terms are sometimes used interchangeably, sometimes not) of a continuum $X$, as I discovered recently when I was looking into this for a paper myself. Consider the following:

1. The condition (W) you attribute to Whyburn: arbitrarily small neighbourhoods whose boundary has only one point. (I hadn't seen this definition before.)
2. The second condition you state, which I will denote (F) [as I have seen it in an article by Fugate, Decomposable chainable continua, Trans. Amer. Math. Soc. 123 (1966)]: sub-continua containing $x$ are linearly ordered by inclusion. (Fugate does refer back to Bing, but IIRC Bing does not make a definition for general continua.)
3. The definition *(A) for arc-like (also called "snake-like") continua given by Ramiro, which is equivalent to (F) for these continua; there are a number of other equivalent formulations for arc-like continua, e.g. using $\epsilon$-maps or the inverse limit characterisation.
4. End-points "in the classical sense": those points $x$ that have an arc starting at $x$, but no two arcs that are disjoint at $x$. Let us call this (E). (I can't remember where this originates, but it is a definition used frequently e.g. when dealing with dendroids, and in particular fans. See e.g. Jacek Nikiel, A characterization of dendroids with uncountably many end-points in the classical sense.
5. Condition (M) (Miller, On unicoherent continua, Transactions AMS 1950) every irreducible subcontinuum of $X$ that contains $x$ is irreducible from $x$ to some point.

There are probably others that I am not aware of; I am not an expert. (EDIT. There are some more in the Gehman reference given by Ramiro.)

Personally I tend to use (F) for "terminal points", and (E) for "end-points". Note that (F) is (as noted by Włodzimierz Holsztyński) not a local property, and seems to make sense most for hereditarily unicoherent continua. In this setting, I seem to recall that it is equivalent to (M) - certainly this is the case for arc-like continua. For the triod, there are three end-points in the sense of (M), (E) and (W) but none in the sense of (F).

Note that any hereditarily indecomposable continuum has every point terminal in the sense of (F), while failing both (W) and (E). The endpoints of the limiting interval of the $\sin(1/x)$-continuum are end-points in the sense of (F) and (E), but not in the sense of (W).

More generally, you see (e.g. using the boundary bumping theorem) that any end-point in the sense of (W) must be a point of local connectivity of $X$, so it may make most sense for Peano continua. To get an example of an end-point of a Peano continuum having a point that is an end-point in the sense of (W) but not in any of the other senses, consider a chain of smaller and smaller closed discs converging to a point; i.e. $$X := \{0\}\cup \bigcup_{n\geq 0} \overline{B}(1/2^n, 1/(3\cdot 2^n)).$$

It would be nice to have a unified usage, but given the long history of the different notions, it is not clear that this is feasible. Likely the best solution is to always make sure the definition used is prominently visible.

Answer 3

The other definition can be localized in more than one way while the following one is perhaps the simplest and natural:

DEFINITION P $\ $ Point $\ x\in X\ $ is an endpoint $\ \Leftarrow:\Rightarrow\ $ for every subcontinua $\ A\ B\ $ of $\ X\ $ such that $\ x\in A\cap B\ $ there exists a neighborhood $\ G\ $ of $\ x\ $ in $\ X\ $ such that one of the sets $\ A\cap G$ and $B\cap G\ $ is contained in the other one.

The P-endpoints and the Whyburn-endpoints are not contained one in the other one, these are two independent notions.