Dense set vs no isolating points

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In summary: A set can be dense without having no isolating points. In summary, a dense subset of a set is a set where every element in the larger set can be approximated by an element in the subset. A set with no isolating points is a set where every element has other elements in its environment. While dense subsets can have no isolating points, it is not a guarantee. For example, the rational numbers are dense in the real numbers but still have isolating points.
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SchroedingersLion
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Differences between two concepts.
Greetings,

could you commend or correct the following:

A dense subset ##X## of a set ##Y## is a set such that in each environment of ##y\in Y##, there is at least one element ##x\in X##. In other words, the elements of ##Y## can be approximated arbitrarily well by elements in ##X##.

A set with no isolating points is a set such that in each environment of ##a\in A##, there are other elements of A.

Are dense subsets automatically sets with no isolating points?
Simple example, ##\mathbb{Q}## is dense in ##\mathbb{R}##, so in each environment of ##x\in \mathbb{R}## there is at least one element ##\in \mathbb{Q}##. But take an ##x\in \mathbb{Q}##. Are there other rationals in each environment of a rational?SL
 
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Yes there are other rationals in each neighborhood of a rational! Given a rational ##q\in \mathbb{Q}##, simply consider ##q+\frac{1}{n}## where ##n## is a sufficiently large natural number (basically this is rewording that ##q+\frac{1}{n}\to q##).

Your definition of dense set seems ok to me. This is equivalent with saying that the closure of ##X## is ##Y## or that ##X## intersects every open set of ##Y##.

A dense set can evidently have isolated points. Consider the space ##Y = [0,1]\cup \{2\}## (with the subspace topology inherited from the usual topology). Then ##X=Y## is dense in itself (trivially) but ##2## is isolated in ##X## since ##\{2\}## is an open set.
 
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Thank you Math_QED,

so while ##\mathbb{Q}## is dense in ##\mathbb{R}## and has no isolating points, it is generally not true that dense subsets have no isolating points.
 
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SchroedingersLion said:
Thank you Math_QED,

so while ##\mathbb{Q}## is dense in ##\mathbb{R}## and has no isolating points, it is generally not true that dense subsets have no isolating points.

Exactly.
 

Related to Dense set vs no isolating points

1. What is a dense set in mathematics?

A dense set in mathematics is a subset of a topological space in which every point in the space can be approximated by points from the subset. In other words, there are no "gaps" or "holes" in the subset.

2. What is the difference between a dense set and a set with no isolating points?

A dense set is a subset of a topological space in which every point can be approximated by points from the subset, while a set with no isolating points is a subset in which every point is a limit point of the subset. In other words, a dense set has no "gaps" while a set with no isolating points has no "isolated" points.

3. How is the concept of dense set vs no isolating points used in real analysis?

In real analysis, the concept of dense set vs no isolating points is used to study the properties of functions and their limits. For example, a function is continuous at a point if and only if the preimage of every open set containing the point is dense. This means that the function is "close" to the point at every point, and there are no isolated points where the function behaves differently.

4. Can a set be both dense and have no isolating points?

Yes, a set can be both dense and have no isolating points. For example, the set of rational numbers is dense in the real numbers, meaning that every real number can be approximated by a rational number. However, the set of rational numbers also has no isolating points, as every rational number is a limit point of the set.

5. How does the concept of dense set vs no isolating points relate to compactness?

A set with no isolating points is always compact, but a dense set may or may not be compact. This is because a set with no isolating points is closed and bounded, which are two properties of compact sets. However, a dense set may not be closed or bounded, and therefore may not be compact.

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