Is there a Closed Unbounded Set?
I just pinned down whether the set of all real numbers from -infinity to +infinity is a closed set or an open set. It is both, a “clopen set”. It is both a closed and an open interval, so we can write either (0, infinity) or (0, infinity] as notation for the positive numbers. The difference is in whether we want to include “infinity” as an object in the set. Mathworld tells me that a closed set is one that contains its limit points (which are *not* necessarily its endpoints). Thus, (0, infinity] does not have to contain infinity itself. An open set is one such that a tiny ball around any point is still entirely within the set, and the set of all numbers from -infinity to +infinity also satisfies that definition. More generally (needing a more general definition too: that a small movement won’t take you out of the set into some “meta-set”), the null set and the whole space are clopen, as Wikipedia tells us.
I was interested in this because I wondered if there could be a closed unbounded set. Yes: an example is [-infinity, +infinity]. But it is also an open set.
September 8th, 2006 at 10:30 am
I think this clopen set,a closed but unbounded set,can just to say the world which is movable,something when we have went through,or we have defination,it is external closed ,but the way is long and we ,even our filial generation can’t illustrate boundlessly,so it is a set which is open.so we can say the world big but really small.
September 8th, 2006 at 12:52 pm
The integers would be another example of a closed unbounded set.
September 18th, 2006 at 4:22 pm
[0, +infinity) is closed and unbounded subset of R, the set of real numbers, and not open, 0 is not an interior point. Also one can consider any line in the plane R^2, it’s closed and unbounded and yet has empty
interior.
September 19th, 2006 at 8:22 am
I like that line in R2 example. It points out the importance of context, and makes sense of the idea that the whole space is both closed and open without tricks of infinity. Consider the line x=3. It is open in the sense that in 1-space every point in it is an interior point, so a ball around any point still contains only points on the line. It is closed in the sense that in 2-space every point is a boundary point— even points like (3,4) that are nowhere near x=infinity.
The x=3 analogizes to any entire space. Looking just at the space, every point is interior, because we can’t move a little and get out of the space. But we can always embed the space in a bigger space, in which case there’s always a small movement that will take us out of the space.
December 18th, 2006 at 3:46 pm
what counts as “closed” or “open” depends on the topology you are talking about. it’s not enough to specify the set of numbers in question. you have to give it structure as well. so, i think the question about unbounded closed sets is ill-defined.
December 19th, 2006 at 11:14 am
True literally, but conventionally when people see numbers they apply whatever topology it is that we use numbers in the everyday sense, rather than treating them as arbitrary objects such as (apple, tree, ball).
Is the question itself ill-defined? It asks whether you can, in any topology, have an unbounded closed set.