Cofinite Topology
Let $$X$$ be a non empty set, and then the collection of subsets of $$X$$ whose compliments are finite along with $$\phi $$(empty set) forms a topology on $$X$$, and is called co-finite topology.
Example:
Let $$X = \left\{ {1,2,3} \right\}$$ with topology $$\tau = \left\{ {\phi ,\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\},\left\{ {1,2} \right\},\left\{ {2,3} \right\},\left\{ {1,3} \right\},X} \right\}$$ is a co–finite topology because the compliments of all the subsets of $$X$$ are finite.
Note:
It may be noted that every infinite set may or may not be a co–finite topology. For this suppose $$X = \mathbb{R}$$ (set of real numbers which is an infinite set) with topology $$\tau = \left\{ {\phi ,\mathbb{R} – \left\{ 1 \right\},\mathbb{R} – \left\{ 2 \right\},\mathbb{R} – \left\{ {1,2} \right\},\mathbb{R}} \right\}$$ is a co–finite topology because the compliments of all the members of topology along with empty set are finite.
Remark:
If $$X$$ is finite, then the topology $$\tau $$ is discrete. For a subset of $$X$$ belongs to $$\tau $$ if and only if it is either empty or its compliment is finite. When $$X$$ is finite, the compliment of each of its subset is finite and therefore, each subset of $$X$$ belongs to $$\tau $$. Hence $$\tau $$ is the discrete topology on $$X$$.
Rashid
January 29 @ 12:01 pm
Disceret and inDisceret topologies are Cofinit or not?
Nadeem Abbas
February 27 @ 3:13 am
I think both are cofinite topologies.
Am I right?
If you have the right answers then please correct me with explanation
Ajay
January 5 @ 10:16 pm
Indiscrete is cofinite but discrete is not cofinite
Geoffrey Kimathi
April 9 @ 7:31 pm
Discrete topology is cofinite,since for a topological space (t) on X which is discrete consist of all possible subsets of X.. Therefore the compliment of all the open sets in X are finite. Also an indiscrete topology is cofinite trivially .