What is the opposite of a "discrete set"?
$\mathcal{N} = \{1, \ldots, p\}$ or $\{1, 2, 3, 4,\ldots\}$ are discrete sets.
So what is the opposite of a discrete set?
There is no such thing as a "continuous set".
I know there is convex set, connected sets as well as path connected sets and their generalizations.
Which concept would most appropriately or generally describe sets (such as $(a,b),[a,b]$ and their products, etc.) where there is no discontinuity when traveling from one point to another? i.e., what is the most general way to describe a "connected set?"
$\endgroup$ 92 Answers
$\begingroup$A crowded space is a $T_1$ space without isolated points. Such need not be connected of course. But it is sort of opposite of discrete in some ways.
I don't think there is a most general concept as you envisage. Path-connected comes close.
$\endgroup$ $\begingroup$A topological space is discrete iff every point is an isolated point iff every singleton is an open set. So you could say the opposite of a discrete space is one in which every nonempty open set contains more than one point: a space which is dense-in-itself.
However, the nomenclature indiscrete space or anti-discrete space usually refers to a different concept, a space where the only open sets are $\emptyset$ and the entire space (i.e. it has the trivial topology). We might think of this type of space as the polar opposite of a discrete space, since in a discrete space, every subset is open.
$\endgroup$
