# Talk:Closed set

## Closeure (topology)

A separate article is titled closure (topology). Should these two articles be merged? -- Michael Hardy, 2003 Aug 12

IMO, no; see my reply on Talk:Closure (topology). -- Toby Bartels 05:13, 25 Sep 2003 (UTC)

Should this be the page to include what it means to be "closed over addition" or the like? (meaning that if x and y are part of the set, x+y must also be part of the set). Fresheneesz 20:21, 17 March 2006 (UTC)

I would say no, as that type of closure is covered by Closure (mathematics). Closure under an operation is not related to whether a set itself is closed in some space. - grubber 21:33, 17 March 2006 (UTC)
Though we can certainly link that in the disambig info at the top. -lethe talk + 06:03, 18 March 2006 (UTC)
I think that's a great idea. - grubber 17:59, 19 March 2006 (UTC)

## Intuition?

I've encountered closedness in various math courses, but never felt fully confident of the intuition behind the definition and so don't feel confident infering anything from the fact that a set is open or closed. My intuition for closedness is that closed sets contain their "edges" For continuous spaces, since no point is arbitrarily close to a given other point, if you select a connected subset of the points in that space, you have to draw the outline somewhere and then you have to decide if that outline does or does not belong to the set of points you are selecting.

I gather that open sets are important in defining topological spaces. That basically each open set is all of the points within a given loop (in the case of a 2D manifold) that the loops tie together the points they contain saying "these points are close together". In my head, I picture an open set as spray of spraypaint—a blob of points with an edge that blurs out to nothing. (Obviously that's not quite the right visualization in that a spray of spraypaint doesn't have a definitive edge whereas an open set does...). I'm rambling because I don't know quite what to ask. How do you visualize closedness and openness and in an intuitive sense, why are these concepts so prevalent in topology and analysis? —Ben FrantzDale (talk) 18:38, 6 December 2008 (UTC)

I'm no mathematician, so I might be mistaking, but I think there is a problem here :

"A subset A of a topological space X is closed in X if and only if every limit of every net of elements of A also belongs to A. In a first-countable space (such as a metric space), it is enough to consider only sequences, instead of all nets"

Shouldn't we say : "it is enough to consider only sequences converging in X" ? If X=${\displaystyle \mathbb {R} }$, if we take the subset A={${\displaystyle x\in \mathbb {R} /x\geq a}$} for some ${\displaystyle a}$ belonging to ${\displaystyle \mathbb {R} }$, we see easily that A's complementary in X is open, so A is closed; but if we take any sequence of elements of A diverging in X, its limit is infinity, and infinity doesn't belong to A, so A is both closed and open... Weird, actually.

Kamixave (talk) 14:31, 10 December 2008 (UTC)

## Is it possible to add an English explanation?

I just commented over in Talk:Open set that I found the explanation there difficult to follow. It was easy compared to this one. Could somebody add an explanation here that doesn't rely on deep math? A minimum would be copying the first two paragraphs from Open set and flipping their sense. I would do it, but I'm not a strong enough math guy to do it safely. See also my comments there about describing the applicability of the terms to non-metric spaces(?).

Jordan Brown (talk) 17:11, 20 March 2009 (UTC)

## Definition in the article

It says "Equivalently, a set is closed if and only if it contains all of its limit points." What about isolated points? Goldencako 16:05, 11 June 2009 (UTC)

## Is this closed?

If one speaks to the set of numbers ending in digits {1,3,7,9} for example, is that a closed set under multiplication, as the result also ends in digits {1,3,7,9}?--Billymac00 (talk) 17:57, 6 December 2009 (UTC)

I think you are confusing closed set and closure. We would say that the set of all numbers whose last digit is 1, 3, 7, or 9 is closed under multiplication (this is the idea of closure under an operation). However, a closed set (this article) is one that is closed under limits, thus e.g. all positive real numbers is closed under multiplication (because positive * positive = positive), but is not a closed set because it contains the numbers {1, 0.1, 0.01, 0.001, ...} but it does not contain their limit, 0. Hope that helps. Cheers, — sligocki (talk) 22:42, 6 December 2009 (UTC)

## Closed set <=> contains its limits

The introduction says that a closed set contains its limits point in a metric space. The first section says that these notions are equivalent in any topological space. I tend to think the introduction is right, but I'm not sure. Is it possible to check and get rid of the ambiguity? 82.243.58.163 (talk) 14:15, 1 April 2010 (UTC)

It's true in any topological space. However, it is much easier to understand for metric spaces. I've edited the article to be more consistent, feel free to add :) Cheers, — sligocki (talk) 04:46, 8 April 2010 (UTC)
Added "complete", as all those limits things can be said if it is a complete metric space. Am I wrong? Marius siuram (talk) 18:40, 4 June 2012 (UTC)

# proof of a closed set

Usually, now to approach this? For example, the set of all bounded continuous non-decreasing functions is closed, but how to prove this? Please don't tell me it is clearly obvious, I can imagine that this is right, but how to prove it? Jackzhp (talk) 22:25, 17 August 2010 (UTC)

The question you ask about closed sets of functions is reasonably complicated. Take a class in Mathematical Analysis or read one of the texts Mathematical analysis#References (Rudin is the standard reference) for a full explanation. Basically, you must first define a metric over the space of functions and then prove that any sequence of functions (which converge), converge to a function in the set. Starting with a simpler example would probably be a better idea. Cheers, — sligocki (talk) 02:33, 19 August 2010 (UTC)

## Cantor Set example

Is it right to say that the Cantor set is unusual for a closed set because "it consists entirely of boundary points and is nowhere dense". Don't many simple examples of closed sets (such as a finite number of points) also have this property? — Preceding unsigned comment added by 98.159.210.6 (talk) 16:30, 11 September 2012 (UTC)