The set E is nowhere dense, ifa)closure of E contains no non-empty open setsb)closure of E contains non-empty open setsc)closure of E contains empty open setd)None of the aboveCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Nowhere dense is a strengthening of the condition "not dense" (every nowhere dense set is not dense, but the converse is false). Another definition of nowhere dense that might be helpful in gaining an intuition is that a set S⊂X is nowhere dense set in X if and only if it is not dense in any non-empty open subset of X (with the subset topology).

For example, Z is nowhere dense in RR because it is its own closure, and it does not contain any open intervals (i.e. there is no (a,b) s.t. (a,b)⊂Z¯=Z An example of a set which is not dense, but which fails to be nowhere dense would be {x∈Q|0

An example of a set which is not closed but is still nowhere dense is {1n|n∈N}. It has one limit point which is not in the set (namely 00), but its closure is still nowhere dense because no open intervals fit within {1n|n∈N}∪{0}.

Answer

Nowhere Dense Sets

A set E is said to be nowhere dense in a topological space if its closure contains no non-empty open sets. In other words, the closure of E consists entirely of boundary points.

Explanation

Let's analyze each option to understand why the correct answer is option A.

a) Closure of E contains no non-empty open sets:

If the closure of E contains no non-empty open sets, it means that every point in the closure of E is a boundary point. In other words, for every point x in the closure of E, every open neighborhood of x contains points both in E and its complement.

Since the closure of E contains no non-empty open sets, it implies that E does not contain any open sets. Hence, every point in E is a boundary point.

b) Closure of E contains non-empty open sets:

If the closure of E contains non-empty open sets, it means that there exist points in the closure of E that are not boundary points. This would imply the existence of open neighborhoods of these points that are entirely contained within E.

If E contains non-empty open sets, it contradicts the definition of a nowhere dense set. Therefore, option B is not the correct answer.

c) Closure of E contains empty open sets:

If the closure of E contains empty open sets, it means that there are no points in the closure of E. This would imply that E is an empty set.

Since nowhere dense sets are non-empty by definition, this contradicts the definition of a nowhere dense set. Therefore, option C is not the correct answer.

d) None of the above:

Since option A is the correct answer, option D is not the correct answer.

In summary:

Option A is the correct answer because a set E is nowhere dense if its closure contains no non-empty open sets. This implies that every point in the closure of E is a boundary point, and E does not contain any open sets.

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