# Category:Denseness

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This category contains results about denseness in the context of Topology.

Definitions specific to this category can be found in Definitions/Denseness.

Let $T = \struct {S, \tau}$ be a topological space.

Let $H \subseteq S$.

Then $H$ is dense-in-itself if and only if it contains no isolated points.

It also contains results about everywhere dense and nowhere dense subsets of topological spaces.

## Subcategories

This category has the following 3 subcategories, out of 3 total.

### C

### I

### N

## Pages in category "Denseness"

The following 47 pages are in this category, out of 47 total.

### C

### D

### E

### I

### N

### O

### R

- Rational Number Space is Dense-in-itself
- Rationals are Dense in Compact Complement Topology
- Rationals are Everywhere Dense in Reals
- Rationals are Everywhere Dense in Reals/Normed Vector Space
- Rationals are Everywhere Dense in Reals/Topology
- Rationals plus Irrational are Everywhere Dense in Irrationals

### S

- Set of Reciprocals of Positive Integers is Nowhere Dense in Reals
- Singleton Set is not Dense-in-itself
- Space is Separable iff Density not greater than Aleph Zero
- Space of Almost-Zero Sequences is Everywhere Dense in 2-Sequence Space
- Subset of Excluded Point Space is not Dense-in-itself
- Subset of Indiscrete Space is Dense-in-itself
- Subset of Indiscrete Space is Everywhere Dense
- Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space
- Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Necessary Condition
- Subset of Normed Vector Space is Everywhere Dense iff Closure is Normed Vector Space/Sufficient Condition
- Subset of Nowhere Dense Subset is Nowhere Dense