How does being an indecomposable continuum imply irreducibility between all pairs of points? A continuum $K$ is called indecomposable if $K$ can not be written as the union of two proper subcontinua $A,B$.
A continuum $K$ is called irreducible between points $x,y\in K$ if there is no proper subcontinua of $K$ that contains both $x$ and $y$.
While reading about indecomposable continua on wikipedia I found the following line:
"An indecomposable continuum is irreducible between any two of its points."
First, is this true? Wiki cites Nadler for this statement, but it sounds suspicious. Second, if this is true wouldn't that imply that indecomposable continua do not contain any nondegenerate proper subcontinua? I suppose a third question would be, is there a name for continua that are irreducible between all pairs of points?
 A: It seems that my issue was with some ambiguous language. The wiki article for indecomposable continuum does contain the statement:
"An indecomposable continuum is irreducible between any two of its points."
I interpreted this to mean the following:
"If $K$ is an indecomposable continuum and $x,y\in K$, then $K$ is irreducible between $x$ and $y$"
Evidently this is not what the statement on the wiki article means. Ostensibly the statement in the article written differently would be:
"If $K$ is indecomposable then there are $x,y\in K$ such that $K$ is irreducible between $x$ and $y$"
The closest thing I can find in Nadler's "Continuum Theory: An Introduction" to my original thought is Corollary 11.19 which reads:
"A continuum $X$ is indecomposable if and only if each point of $X$ is a point of irreducibility of $X$"
Where a point of irreducibility is simply a point $x$ for which there is another point $y$ such that $X$ is irreducible between $x$ and $y$.
A: Alessandro Codenotti pointed out in the comments that a space which is irreducible between all pairs of points cannot be a continuum (every continuum has a proper subcontinuum with more than one point by the boundary bumping theorem).
However, there are connected completely metrizable spaces which are irreducible between every two of their points. The term used for such spaces in the literature is "widely-connected". See https://arxiv.org/pdf/1608.00292.pdf
