Talk:Open set

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." ?

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

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)

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)

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

Exactly. Take the example of the open interal $(0,1)$. Choose a point $x\in (0,1)$. Then the open interval $(x/2,(1+x)/2)$ is always contained in $(0,1)$. Since x > 0 it follows that x/2 > 0 and since x < 1 it follows that (1+x)/2 < 1. The closer x gets to the points 0 and 1 the smaller the allowable wiggles. But no matter how close x is to either 0 or 1 you can always find a neighbourhood containing x which is contained in $(0,1)$ (just let the neighbourhood go half way up to the boundary points. Dharma6662000 (talk) 00:56, 26 August 2008 (UTC)

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)

I agree that many people would not be happy, but don't the Wikipedia guidelines say that the exact definition must be in the first few lines? Dharma6662000 (talk) 01:09, 26 August 2008 (UTC)

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)

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://wiki.riteme.site/wiki/Talk:Lebesgue_measure#Borel_vs._lebesgue Jackzhp 17:28, 4 October 2006 (UTC)

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)

Relationship

I'd like to add the following text to the Note paragraph:

a set can be both (open & closed). If a set is not open, then it is not necessarily to be closed. If a set is not closed, it is not necessarily to be an open set.

Jackzhp (talk) 20:26, 14 September 2008 (UTC)

Is it possible to add an English explanation? 50% :-)

I'm not a serious math guy, but I think I know more than the average non-mathematician, and I found this explanation very difficult to follow. I think part of the problem is that most of the sets I've worked with involve finite, unorderable members - {Apple, Banana, Orange}, {UNIX, Windows, MacOS}. Even when those sets are infinite (e.g., assuming the universe is infinite, the set of all particles in the universe), the sets I've worked with involve unorderable members and the concepts of "move" and "direction" and "distance" are not well-defined. Digging further, I think I'm referring to sets that are not in metric spaces. Is there a definition of "open set" that makes sense for sets with non-metric-space sets?

Jordan Brown (talk) 17:00, 20 March 2009 (UTC)

The article is a bit unclear on this I agree. Topological openness in general isn't mentioned until the middle of the article. However trying to explain this at the start requires an explanation of topology, which assures that the article will not be accessible to ordinary readers.

The intro dodges this by trying to restrict the discussion of open sets to metric spaces. Perhaps the intro could add something like how metric openness is a kind of topological openness which in principle is defined by certain conditions unrelated to notions of distance, and how metric openness in fact is one particular way of expressing it. 118.90.84.3 (talk) 08:40, 4 April 2009 (UTC)

Metric spaces are a subset of topological spaces. What that means with regards to your question is that the most basic definition of an open set applies to more than metric spaces. You can turn any set into a topological space by defining which subsets are considered open keeping certain rules in mind. This works also for finite sets.--91.11.119.11 (talk) 11:34, 12 September 2011 (UTC)

Rewrite of the lead

Although the current lead is far from perfect, it is somewhat an improvement of the original lead. In particular, this lead focuses more on the ubiquity of open sets in various branches of mathematics, as opposed to an "intuitve discussion." However, I have preserved the old lead and merged it into a section called "informal discussion" as I am aware that the accessibility of this article was disputed. Within the next 24 hours, I aim to improve the intuition section and thus the quality of the article. Any suggestions and criticisms of the new lead are welcome because I am aware of possible repitition within it. However, I feel that this article deserves more attention and thus my enthusiasm to improve it. --PST 09:35, 7 June 2009 (UTC)

Geometric definition of open set: an example

Let S be the set of all rational numbers x such that 0 < x < 1. Is this an open subset of R¹? TomyDuby (talk) 03:16, 7 September 2010 (UTC)

No. Any open ball in R¹ contains an irrational, so no set containing only rationals can be open. Paul August ☎ 12:15, 7 September 2010 (UTC)

Paul, Thanks for clarifying this. Then the definition of the "internal point" in Interior (topology) needs clarification in the sense I introduced in this article.TomyDuby (talk) 19:12, 7 September 2010 (UTC)

You mean "Interior point", but I'm not following you. What exactly needs clarification? Paul August ☎ 19:21, 7 September 2010 (UTC)

The current definition of interior point in Interior (topology) says: "If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is contained in S." From this definition it is clear that the open ball is a subset of the Euclidean space. I hope that this answers your question. This lack of clarity brought me to my original question. TomyDuby (talk) 18:00, 8 September 2010 (UTC)

Geometric definition

The geometric definition is,

A point set in Rⁿ is called open when every point P of the set is an interior point.

And the defintion of interior point says,

If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is contained in S.

so... to understand open set, I have to first know what an open ball is? Maybe we could have a non-circular definition. 0¹⁸ (talk) 19:44, 8 September 2010 (UTC)

If you look at open ball you find the definition in terms of the metric on Rⁿ. These open balls then define a basis from which all open sets can be constructed and then define the topology on R ⁿ. So not strictly circular, but could be made somewhat better. I'm not convinced we need to have the Geometric section at all the Euclidean space is more precise.--Salix (talk): 22:14, 8 September 2010 (UTC)

So you need a basis to have an open set? 0¹⁸ (talk) 23:52, 8 September 2010 (UTC)

In point-set topology the topology of a space is defined by the collection, T, of all open sets. A Base (topology) is a way of defining T as every open set in T can be written as a union of elements of the basis. In general PST then you can choose a definition of an open set which may or may not involve a basis, their might be some spaces where the basis is identical to T. In Euclidean space the topology is specified from the metric on the space which defines a basis. --Salix (talk): 05:36, 9 September 2010 (UTC)

Intuitive discussion

I'm confused by the phrasing of "Intuitively speaking..." This is followed by a reference to "a more intuitive discussion". If you are already speaking intuitively just carry on. As written, I read the meaning to be "I'm going to be talking intuitively for a while. And then I'm going to be talking intuitively." So is the motivation section really more intuitive? If so it should be referenced after the completion of the intuitive explanation, not right in the middle, and stated something like, "To continue with the less formal discussion, see ...".

I hesitate to make the change myself because I'm not sure what the intent of the sentence is (is it intuitive now? or later? or both?), so I'll leave this comment here. If no one either changes it or tells me I'm wrong in a few days, I'll reword it.

Paul D. Anderson (talk) 01:07, 19 September 2010 (UTC)

Clarification

I think the beginning of the article could use a little clarification. My understanding of sets is somewhat basic, but I'm rather confused by the use of the term "direction." It seems like an arbitrary relationship that doesn't have an analogue for every set. If I compose a set containing the elements {happy, purple, fast, symmetrical, small} or even {true, false}, there is no "directionality," so how do you relate that to an open/closed set?

It seems like there has to be a specific type of relationship between the elements of a set for open/closed to apply. "Direction" seems like it could apply to any set where the elements are non-commutative, but perhaps it is only relevant for metric sets? It would be nice if the article could reflect and specify that in some way. 71.59.61.206 (talk) 22:50, 16 January 2011 (UTC)

I rewrote the first paragraph to talk about distance, which is defined (since the setting is a metric space), rather than "moving in a direction" which need not be defined, and if defined, need not be possible (e.g. an isolated point in a metric space is open as a singleton set, although you can't "move it around" precisely because it's isolated). ChalkboardCowboy (talk) 17:55, 24 June 2012 (UTC)

Who Will Actually Benefit From Reading This Page

IMHO: I wanted to see a crisp, simple, valid, and meaningful definition of an open set right up front; the very first thing I expect to see when the page pops up. That's why I called up Wikipedia at the present time. I didn't come here to learn that an open set is important to many areas of mathematics, like point-set topology and metric topology. . . And I certainly don't want to see mathematics watered down in a way that some highly paid math denying Republican hack writer believes will make it become accessible to idiots, where everything is written 'intuitively'. WTFIGOAW?

And would someone please explain the sentence after next? "Intuitively, an open set provides a method to distinguish two points. For example, if about one point in a topological space there exists an open set not containing another (distinct) point, the two points are referred to as topologically distinguishable." Since when did one existing point become distinguishable from another point that does not exist?

Langing (talk) 23:56, 10 June 2011 (UTC)

The topological definition of openness is much simpler than explained here. A set is called open with respect to a given topology iff it is an element of the topology of X. As a corollary, a set is called closed with respect to a given topology iff the complement is in the topology of X. -- 139.18.248.20 (talk) 11:15, 18 April 2012 (UTC)

"More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology."

Another example; that's among the worst sentences I have ever seen. Both the collection and set are "given"? What does that even mean. Is everything after "a collection..." referring to what a set is? If not why is there a comma after generally? I would make this article make sense if I could figure out what the hell it's trying to say. 184.67.37.222 (talk) 02:02, 7 April 2022 (UTC)

First sentence

I have a big beef with the first sentence:

a set $U$ is called an open set if it is a neighborhood of every point $x\in U$.

First the definition is unhelpful, as a reader who doesn't know what an open set is is unlikely to know what a neighborhood is.

So our reader will we forced to check the neighborhood article and will discover that a neighborhood of a point is a set which contains an open set containing that point. The definition therefore translates to:

A set $U$ is open iff, for every point $x\in U$, there is an open set in $U$ containing $x$.

Now of course if $U$ is open then for any $x\in U$, $U$ is an open set in $U$ containing $x$ so this is tautological. In the other direction we have that

If for each $x$ in a set $U$ there is an open set in $U$ containing $x$, then $U$ is open.

And that is perfectly useless.

I'll much prefer something like:

An open set is a set with no boundary point.

At least this has a reasonably intuitive meaning and corresponds to the illustration at the beginning. Of course if you dig deeper it is still circular but nevertheless helpful.

Suggestions are welcome but the current sentence clearly doesn't work. Bomazi (talk) 14:25, 28 May 2012 (UTC)

I rewrote the first paragraph following your suggestion, although instead of "with no boundary points" I said "does not contain any of its boundary points". The former could be confusing since the boundary of an open set is defined, so one might say the open set "has" boundary points.

I also removed the part about being able to "move" the point some distance in any "direction" (which is not just fuzzy but also inaccurate--consider the open set $\{2\}$ in the metric space $[0,1]\cup \{2\}$). I hope the slight reduction in clarity is worth the improvement in correctness. ChalkboardCowboy (talk) 17:53, 24 June 2012 (UTC)

Thanks for the reformulation, I was sloppy. Bomazi (talk) 23:17, 24 June 2012 (UTC)

Circular definitions: open set <-> neighborhood

In this article, an open set in a topological space is defined in terms of the concept of "neighborhood". To my horror, when I click on "neighborhood" I find that it is defined in terms of the concept of "open set"! Gabn1 (talk) 11:10, 4 February 2013 (UTC)

I too noticed this. --193.219.42.50 (talk) 08:43, 5 July 2013 (UTC)

Exactly. One year passed, and there's still circular definition. Any solution? Adam Stankiewicz (talk) 18:44, 11 May 2014 (UTC)

Fixed. Brirush (talk) 19:34, 17 November 2014 (UTC)

Notes and Caution one more example for '"Open" is defined relative to a particular topology'

Can we add one more example where an open disc in the article's first picture would NOT be considered open if we consider $\mathbb {R} ^{3}$ as the space (rather than $\mathbb {R} ^{2}$) since now epsilon "balls" would have to be considered? This may be more easy to understand than the example with rational numbers. 103.21.125.77 (talk) 09:04, 20 August 2015 (UTC)

Standard Definition of an Open Set Somehow Omitted

An open set is a set which does not contain its boundary.

That is the standard topological definition. How can the article miss this? Why does the article beat around the bush and speak in mysteries instead?

I suggest to consider a serious answer as to why. It's not a mistake.

Find a source that you want to use and include the standard definition. TheodoricStier (talk) 16:21, 30 November 2023 (UTC)

One cannot have a definition of a boundary without a previous definition of open sets. So your assertion cannot be a definition. Moreover, your assertion is wrong, the correct assertion being : "a set is open if and only if it does not contain any point of the boundary". However this assertion is of low interest, as, for computing the boundary, one generally requires to know whether the set is open. D.Lazard (talk) 08:50, 3 December 2023 (UTC)

The article needs to offer at least one formal definition. Various authors do it either way. The definition of openness in terms of a boundary is common. You cannot possibly mean to portray a formal definition of a mathematical entity as being of low interest to the matter.

TheodoricStier (talk) 22:08, 20 March 2024 (UTC)

The definition you gave is wrong anyway. You gave the definition of a non-closed set, not an open one. TheodoricStier (talk) 22:09, 20 March 2024 (UTC)

You must be more specific. There are two definitions in the lead and three in section § Definitions. All these definitions are compatible, in the sense that when two definitions can be used, they are equivalent. So, you must say which definition is different from the common usage (a definition being an arbitrary choice, it cannot be wrong), and what it should be exactly. D.Lazard (talk) 08:07, 21 March 2024 (UTC)

"Closed set" term used before defined.

I think the definition of "closed set" should be moved up from the "Properties" section so it precedes the "Clopen sets and non-open and/or non-closed sets" section. Mnewbold (talk) 21:22, 19 October 2024 (UTC)

Section § Special types of open sets was misplaced and I moved it down. Also, I moved the content of section "Notes and cautions" into section § Properties, and I removed the heading "Notes and cautions" and its unique sub-heading. D.Lazard (talk) 08:50, 20 October 2024 (UTC)