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Today, I learnt that the empty set is considered as an open region and as a closed region. The argument used was that since the empty set is not "not an open region", it is considered to be an 'open region'. Similarly, since the empty set is not "not a closed region", it is considered to be a 'closed region'.

According to the prescribed textbook for my General Mathematics course, the 14th Edition of Thomas' Calculus (ISBN: 978-93-530-6041-1), the definitions of an interior point, a boundary point, a closed region and an open region are as follows:

A point $(x_0, y_0)$ in a region (set) $R$ in the $xy$-plane is an interior point of $R$ if it is the center of a disk of positive radius that lies entirely in $R$ (Figure 14.2). A point $(x_0, y_0)$ is a boundary point of $R$ if every disk centered at $(x_0, y_0)$ contains points that lie outside of $R$ as well as points that lie in $R$. (The boundary point itself need not belong to $R$.)

The interior points of a region, as a set, make up the interior of the region. The region's boundary points make up its boundary. A region is open if it consists entirely of interior points. A region is closed if it contains all its boundary points (Figure 14.3).

This definition implies that there exists an element in the set which can be referred to as a point. In the case of the empty set, there are no such elements and thus no 'points'. Hence, there is no merit in defining an 'interior point' or a 'boundary point' as the definitions for the same reference a 'point', which in this case doesn't exist. Similarly, the definitions for an open region and a closed region imply that 'interior points' and 'boundary points', respectively, are defined.

However, as seen in the case of the empty set, there are no 'points'. As per my understanding, it is thus, not possible to use the aforementioned definitions to conclude that an empty set is an 'open region' or is a 'closed region'.

How is it then that an empty set can be said to be an 'open region' and a 'closed region'? If there is any flaw in my reasoning, please do let me know, so I can better understand this.

Thanks.

Sammy Black
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jaytau
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    "This definition implies that there exists an element in the set which can be referred to as a point". -- Does it? – Vercassivelaunos Dec 11 '21 at 07:30
  • Every point is trivially interior, so the set is open. Its complement is the whole space, which is open, therefore it is closed. – Zanzag Dec 11 '21 at 07:36
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    The statement "If $x\in A$ then $______$" is true if the statement "$x\in A$" is false. – David P Dec 11 '21 at 07:36
  • For empty set all its points are interior, because, simply, they do not exist. – zkutch Dec 11 '21 at 07:45

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