Talk:Open set

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[edit] Rigorous Definition

Shouldn't it say somewhere "A set E is said to be open in a (metric) space X if every point of E is an interior point of E." ?

[edit] Example

give example of a set which is both open and closed in euclidean spaces both the empty set and the whole space itself are simultaneously open and closed.

Why is e used instead of ε? --anon

[edit] Typo?

Should the "rational" and "real" be switched in the second paragraph?

As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1.

Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers.

I think the second paragraph is good as it stands. When nothing is mentioned about the surrounding space it is assumed to be the usual topology on the real line.
Does this answer your question? Oleg Alexandrov 19:23, 18 Mar 2005 (UTC)

"The set of rational numbers between 0 and 1 (exclusive) is not open in the real numbers." I find this unclear. Nossac 23:45, 15 January 2006 (UTC)

Nobody said that the set of rational numbers between 0 and 1 is open in the real numbers. That is not in the article. Oleg Alexandrov (talk) 02:28, 16 January 2006 (UTC)
The sentence is "For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers." I read the 'it' after the comma as referring to 'the set of rational numbers between 0 and 1 (exclusive)', hence the meaning is as in my comment above. If I am misreading the sentence, then this is my reason for stating that it is unclear! Nossac 10:23, 16 January 2006 (UTC)
If you "wiggle" a little bit, you intersect with the irrationals. May be the article should do a little hand holding and state that? 127 16:59, 16 January 2006 (UTC)
I see, so you are allowed to 'wiggle' in the real line, even though the set we are dealing with is limited to the rationals?" Nossac 17:23, 16 January 2006 (UTC)
Well yes, of course. If you "wiggle" and you intersect the complement, then its not open. You can not determine what the complement is without knowing what other set your set resides in. If your set is not in another set, then its both open and closed. 127 18:02, 16 January 2006 (UTC)
If you want to know whether a set is open in the rationals, then you forget altogether about irrationals, they are out of the universe for the moment. And then yes, if you wiggle a bit the set of rationals between 0 and 1 (exclusive) you are still left in the rationals between 0 and 1. Oleg Alexandrov (talk) 23:49, 16 January 2006 (UTC)

[edit] Intuitivity

I've always pictured open and closed sets as drawn on paper with a pencil. The boundary is drawn by pressing the pencil in a normal writing angle at the paper, making a clearly visible, narrow, black line. The interior, however, is drawn by holding the pencil almost parallel to the paper and shading the area, making it a fuzzy gray. Closed sets have boundaries around the interior, open sets don't.

This has the added bonus of being able to visualise a separation of a space into an open set and a closed one. The line in the middle has to belong to exactly one set. — JIP | Talk 19:38, 18 Mar 2005 (UTC)

wtf is a 'wiggle' in the context of topology? --138.25.80.124 07:44, 8 August 2006 (UTC)

That is an intuitive real-world meaning, not topological meaning.
OK, topologically, the current point has wiggle room in a set, if it is contained in a neighborhood which is contained in that set. Oleg Alexandrov (talk) 15:27, 8 August 2006 (UTC)

[edit] Intervals

I'm quite confused by the definition of an open set. The text says:

"As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if you take x = 1 and wiggle a tiny bit in the positive direction, you will be outside of (0,1]."

What I have issue with is the statement that the interval (0,1) is open but (0,1] is not. From the description I conclude that if (0,1] is not open then neither is (0,1). The reason I say this is that just like in the case of the set (0,1] if you take x as 1 and wiggle it a bit and get a number outside of the set you can do a similar thing with the set (0,1). Suppose we take an x = 0.0000000000000000000000000001 in (0,1) and wiggle it a bit and get 0 well 0 is outside of the set (0,1). Therefore, (0,1) is not open. At least that's my reading of it. Is my understanding correct or am I making a mistake somewhere? (Unsigned comment by User:SachinKainth)

I think the "intuitive definition" is not very well expressed here. What it is trying to say is that given any x in (0, 1) no matter how close to 0, or 1, you can always "move" it both to the right and to the left some distance and still remain inside (0, 1). For example you can always move it half the distance from x to 0 to the left, and half the distance x to 1 to the right. Notice that it says wiggle "a little bit (but not too much)", It your example you "wiggled" x "too much". In the case of (0, 1], for x = 1, moving x any distance to the right will move it outside the interval (0, 1]. — Paul August ☎ 13:53, August 17, 2005 (UTC)
Another way to put it is however close you go to the edge of the set, there are always points between you and the outside of the set. 127


[edit] Language

In keeping with the spirit of this reference website, it would probably be useful if the true, mathematical definition of an open set appeared at the beginning of the article. All the talk of "wiggling" seems more apt to appear in an intuitive clarification that would follow. --anon

Many people would not be happy with the mathematical definition of an open set appearing at the very top, and I would understand why. It would look too intimidating for some people to even bother read the next paragraph I think. Oleg Alexandrov (talk) 02:48, 10 February 2006 (UTC)

[edit] Request for clarification

the 3rd paragraph, the 2nd sentence: For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers.

I just don't understand it. (0,1) is open in real number? (0,1) is not open in rational numbers? --unsigned

Not quite. This was sort of discussed above. The bit you're missing is that *the set of rational numbers* (0,1) is open in the rationals but not open in the real numbers. Consider x from the rational numbers (0,1). It should be clear that there is another rational number arbitrarily close to x on either side and hence it is open (by the "wiggle" argument used in the text). But now consider x from the rational numbers (0,1) in the context of the real numbers. Because the rational numbers are strictly contained in the real numbers, from the point of view of the real numbers the set of rational numbers (0,1) is missing a large number of points that are contained in the set of real numbers (0,1). So now if you have the x an element of the set of rationals (0,1) but consider that set to be contained in the "larger" set of reals, when you "wiggle" x by a small amount you can end up with a real number that is irrational, so the set is not open.
Said slightly differently: It's clear that the open interval (0,1) of rationals is a subset of something (at least as we're discussing it here). For the present purposes we can consider it to be either a subset of the reals or of the rationals. The important thing to keep in mind is that regardless of which set we imagine it to be a subset of, it still contains only the rational numbers in that interval. When considered as a subset of the rationals, it contains every part of that set by definition. When considered as a subset of the reals, it still has only the rationals and is thus missing all the irrationals in the interval, so it is not open. (I realize these are not the most technical arguments, but I hope they at least give some sense as to why it is true). 128.197.81.181 17:42, 8 June 2006 (UTC)

If 128.197.81.181 doesn't object, I'm going to put some of his or her above comments into the article. Foxjwill 01:28, 26 May 2007 (UTC)

[edit] open set on R

open interval (a, b} a<b is an open subset of R. In my textbook (A Probability Path by Sidney I. Resnick), Borel set is said to be: B(R)
=sigma ((a,b), -infinite<=a<=b<=+infinite)
=sigma ([a,b), -infinite<a<=b<=+infinite)
=sigma ([a,b], -infinite<a<=b<+infinite)
=sigma ((-infinite,x], x belongs to R)
=sigma (open subsets of R)

So, it seems that open subsets of R are not necessary to be (a, b). Is there anybody can give me an example of a subset of R which is not an open interval? Thanks. Jackzhp 21:35, 29 September 2006 (UTC)

Not quite sure what you are asking for here. If you want a subset of R which is not open, then any closed set such as a singleton set {x} will do. If you meant 'an open subset of R which is not an interval', then how about a set like the union of (1,2) and (2,3)? It is not difficult to prove that any open set in R is actually a union of a countable collection of open intervals. Madmath789 21:52, 29 September 2006 (UTC)
The answer to this question might be related to the relation among Borel set, Lebesgue measurable set, and the power set of R. http://en.wikipedia.org/wiki/Talk:Lebesgue_measure#Borel_vs._lebesgue Jackzhp 17:28, 4 October 2006 (UTC)

[edit] Rework of the article

The article seems very hard to understand if not incoherent. I have reworded the beginning and purged the "wiggle" out as it only added to further confusion. Removed the section on open manifold as it is unrelated. Created a Note section and moved some remarks made at the beginning of the article into that new section. I will be formalising the mathematical definitions in the future. --Tchakra 01:20, 29 August 2007 (UTC)

After remarks from Oleg, i moved back the example to the introduction. In addition, i have added a "proprieties" section and reworded some paragraphs. --Tchakra 15:02, 29 August 2007 (UTC)