## What is usual topology?

The usual topology on the reals is defined as the topology generated by the basis consisting of all open intervals in R with the usual ordering.

### What is discrete topological space?

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set.

**Is Cofinite topology separable?**

topology on X. Z =⋂{C ⊆ R : Z ⊆ C, C is closed in R with the given topology} = R, since the only closed set in R containing Z is R. Thus, R with finite complement topology is separable because it contains a countable dense subset.

**Is 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 discrete and indiscrete topology?

set X is called the discrete topology on X, and the collection consisting only of the empty set and X itself forms the indiscrete, or trivial, topology on X. A given topological space gives rise to other related topological spaces.

### Why is it called discrete topology?

If you think of topologies that can arise from metrics, the discrete topology arises from metrics such as d(x,y)={0x=y1x≠y. This metric “shatters” the points X, isolating each one within its own unit ball. Because points are isolated in this way, it makes sense to call the space “discrete”.

**Is the cofinite topology second countable?**

(xxi) The space N with the cofinite topology is second countable. True. Hint: There are only countably many cofinite subsets of N. (xxii) Let X be any set endowed with the cofinite topology.

**What is the cocountable topology of a set?**

The cocountable topology or countable complement topology on any set X consists of the empty set and all cocountable subsets of X, that is all sets whose complement in X is countable. It follows that the only closed subsets are X and the countable subsets of X .

## What is the difference between discrete and cocountable topology?

However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique. Since compact sets in X are finite subsets, all compact subsets are closed, another condition usually related to Hausdorff separation axiom. The cocountable topology on a countable set is the discrete topology.

### Is uncountable topology Hausdorff?

If X is an uncountable set, any two open sets intersect, hence the space is not Hausdorff. However, in the cocountable topology all convergent sequences are eventually constant, so limits are unique.