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Problem

Let $(X,d)$ be a metric space where $X$ is a non-empty set. Is the empty set an open ball in $X$?

I think that it is true because if $X=\mathbb{R}$ with the usual metric then for all $a\in\mathbb{R}$ we can say that the set $(a,a)$ is an open interval in $\mathbb{R}$.

But I can't devise a proof of this. Can anyone help?

By the way I am using the following definition of an open ball,

Open Ball (in a Metric Space)

Let $(X, d)$ be a metric space and let $r\in\mathbb{R}^+$. Then the set, $B_d(x, r) := \{y \in X : d(x, y) < r\}$ will be said to be the open ball of radius $r$ centered at $x$ in the metric space $(X, d)$.

Community
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  • Well, it can be considered as the ball of radius $0$ I guess. Is it that important? – Giuseppe Negro Feb 03 '16 at 13:55
  • @GiuseppeNegro: Actually, I had a debate with one of my professor regarding this issue where he said that $\emptyset$ is not an open ball. –  Feb 03 '16 at 13:57
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    @Watson: My question asks for whether $\emptyset$ is an Open Ball, not Open Set. –  Feb 03 '16 at 13:58
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    An open ball must be centered at some point. – Jimmy R. Feb 03 '16 at 14:00
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    No? I mean, it all depends on how you define Open Ball. In the traditional formulations I have seen the ball is required to have a greater than 0 radius In any way, forget about trying to get a formal proof of the fact: it is a matter of definition. – Jsevillamol Feb 03 '16 at 14:02
  • To @JimmyR. 's point, most authors will require also radius $r\gt 0$ about the center point of a ball. See for example Open balls in general metric spaces. – hardmath Feb 03 '16 at 14:24
  • @VikrantDesai: I also thought so but couldn't prove it. –  Feb 03 '16 at 14:59
  • @hardmath Yes, indeed. So, I think according to most common definitions, I am on the "no" side. – Jimmy R. Feb 03 '16 at 15:00
  • The definition you are giving distinctly states that the radius can not be zero. So d(x,x) = 0 < r for all possible r and all possible x. So x $\in$ B_r(x) for all possible open balls. So no open ball is empty and the empty set is not an open ball. – fleablood Feb 04 '16 at 06:32
  • If (a,a) is an open interval than you are using a different definition than the one you quoted. In the definition you quoted what is the cent of (a,a) and what is the radius? Of course if there is a different definition of open ball, then it could work. By my definition (a,a) = empty set is an open interval but by my definitions I have no reason to assume every open interval is an open ball. – fleablood Feb 04 '16 at 06:46

2 Answers2

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It is not a matter of "thinking", "considering" or "debating". Your professor perhaps has given a definition of open ball. Or, at least, he must have assumed some definition.
That definition should specify if the radius of the ball must be a positive number or null radii are allowed.

From my experience, most books that include a definition of open ball say that the radius must be positive; in this case, the empty set is not a ball in any metric space, since the center must belong to the ball.

In any case, topologic and metric properties are not affected in any way.

ajotatxe
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    But then how $\emptyset$ is an open set? I mean $\emptyset$ should be an open set (by definition, i.e., the one which I am using) if every element of it is the center of an open ball. How then is this "vacuously true"? –  Feb 03 '16 at 14:38
  • See this answer of mine for a help on vacuously true statements. – Crostul Feb 03 '16 at 16:26
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    It's vaccuously true because the empty set doesn't have any points. So every point is the center of an open ball-- all zero of them!. Also every point is a flying purple pig so if you have a definition of glipple = "a set in which every element is a flying purple pig" the the empty said is vacuously glipple. An open ball is defined to be centered at a point and defined to be an open radius. Such a set is never empty. – fleablood Feb 04 '16 at 06:38
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Actually there are two related questions:

  1. Given the definition you cite, is the empty set an open ball?

    This already has been answered by ajotatxe: The definition you cited explicitly excluded radius $0$, and open balls of positive radius are not empty.

  2. Should a reasonable definition include the empty set as open ball?

    When looking at the definition in isolation, it seems arbitrary to require $r>0$. However the introduction of the open ball has a specific purpose: Namely to allow the more general definition of an open set as a set that contains an open ball around any of its points. If you'd include the empty set in the set of open balls, then this definition would result in every set to be open (as any set contains the empty set), which clearly is not wanted.

    Now we could fix that be requiring that every point has a non-empty open ball around each point, but given that open balls are defined specifically for this purpose, the reasonable thing to do is to include that condition in the definition.

    Also note that by excluding the empty set, we always have $x\in B_d(x,r)$, thus any open ball around $x$ is a neighbourhood of $x$. Again, we'd have to make an exception for the empty set if we'd include it in the definition of open balls.

celtschk
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