- #1

SchroedingersLion

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- TL;DR Summary
- 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

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