# What is a dense subset of R?

Table of Contents

## What is a dense subset of R?

Definition 78 (Dense) A subset S of R is said to be dense in R if between any two real numbers there exists an element of S. Another way to think of this is that S is dense in R if for any real numbers a and b such that a

### What is dense set in topology?

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A; that is, the closure of A constitutes the whole set X. The density of a topological space X is the least cardinality of a dense subset of X.

**Is R with the Cofinite topology compact?**

So we can find some finitely many open sets from U to cover this remainder. Therefore every open cover has a finite subcover. So A is compact, but A is arbitrary, so every subset of R is compact.

**Which subsets of N are closed in the Cofinite topology?**

The set of all subsets of X that are either finite or cofinite forms a Boolean algebra, i.e., it is closed under the operations of union, intersection, and complementation.

## Is Z dense in R?

(a) Z is dense in R . that is the case, then there are two consecutive integers n and n + 1 in ( a, b ), so any rational number in the interval ( n, n + 1) is an element of Q \ Z in the interval ( a, b ).

### Is Q is dense in R?

Theorem (Q is dense in R). Combining these facts, it follows that for every x, y ∈ R such that x

**Is R dense in R?**

And of course R itself is dense in R. Another example of a dense subset of R is R∖Z, the set of real numbers that are not integers: you can easily prove that if a

**Is N dense in R?**

Because (0,1) is an open set, it intersects any dense subset of R. This implies that N is not dense in R, as it does not intersect (0,1).

## Is cofinite topology sequentially compact?

All spaces that have the cofinite topology are sequentially compact.

### Is the cofinite topology hausdorff?

An infinite set with the cofinite topology is not Hausdorff. In fact, any two non-empty open subsets O1,O2 in the cofinite topology on X are complements of finite subsets.

**What is cofinite topology on R?**

In the cofinite topology, ‘A is closed’ means ‘A is finite or is R’ and ‘A is open’ means ‘A has finite complement or is empty’. In particular… If A⊆R is infinite then the only closed set containing A is R, and hence clA=R. In particular, if A is infinite, every point of R−A is a limit point of A.

**Is R with cofinite topology connected?**

If by the cofinite topology you mean the finite complement topology, then the only closed sets are finite sets and R itself. Thus every infinite subset of R is dense.