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.