Which of the following is correct?a)A non-empty finite set is not a nbd of any point.b)Every point of a non-empty finite set is an interior point.c)Some points of a non-empty finite set is an interior pointd)A non-empty finite set is a nbd of each of its points.Correct answer is option 'A'. Can you explain this answer?

Question

Answer

A set can be a nbd of a point if it contains an open interval containing the point. Since, an interval necessarily contains an infinite number of points. Therefore in order that a set be a nbd of a point it necessarily contain an infinity of points. Thus, a finite set cannot be a nbd of any points. Hence, no points of finite non-empty set is an interior point.

Answer

A non-empty finite set is not a neighborhood of any point.

A neighborhood, or 'nbd' for short, of a point in a set is a subset of that set that contains an open interval around the point. In other words, a neighborhood of a point includes all points that are close enough to the given point.

Explanation:

To understand why option 'A' is correct, let's consider the properties of a non-empty finite set.

• A non-empty finite set contains a finite number of elements, meaning there is a specific countable number of distinct points in the set.

• Since the set is finite, there is a maximum distance between any two points in the set.

Now, let's consider a point within this non-empty finite set.

• Since the set is finite and has a maximum distance between any two points, it is not possible to find an open interval around a specific point within the set that contains only points from the set.

• Any open interval around a point within the set will necessarily contain points from outside the set. Therefore, the set cannot be a neighborhood of any of its points because it cannot satisfy the condition of containing an open interval around the point.

Example:

Let's consider a simple example to illustrate this concept. Suppose we have a non-empty finite set {1, 2, 3, 4, 5}.

• If we take the point 3 within this set, we cannot find an open interval around 3 that contains only points from the set {1, 2, 3, 4, 5}. Any open interval around 3 will include points outside the set, such as 2.5 or 3.5.

• This demonstrates that the set {1, 2, 3, 4, 5} is not a neighborhood of the point 3.

Therefore, option 'A' is correct: A non-empty finite set is not a neighborhood of any point.

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