The Co-finite Topology on X is discrete if and only if X is finite[Theorem and Proof]

The Co-finite Topology on X is discrete if and only if X is finite

This is a very important theorem of the cofinite topology, understanding theorems and proofs should not be a problem at this stage of mathematics, because if you have been studying mathematics up to the level of general topology, then you should be conversant with theorems and proofs.

So i will try as much as possible to make the proof of this theorem understandable.

Theorem: Let $\mathcal{T}$ be the finite-closed topology on a set X. If $\mathcal{T}$ is also the discrete topology, prove that the set $X$ is finite.

Proof: Let $X$ be finite, then we shall prove the co-finite topology on $X$ is a discrete topology.

The assumptions are obtained from the statement of the theorem.

Now if $X$ is finite, so complement of each subset of $X$ is finite, let $\mathcal{T}$ be the cofinite topology on $X$. Then $\mathcal{T}$ will contain all the subsets of $X$, and hence, we say $\mathcal{T}$ is also a discrete topology on $X$.

We can also prove this conversely, we let the cofinite topology on $X$ be a discrete topology. Then we are suppose to prove that $X$ is finite, but since we are proving by contradiction now, we let $X$ be infinite, then we construct a new set $A$ to be a finite subset of $X$,then $A^c$ is infinite, let $\mathcal{T}$ be a cofinite topology on $X$, then $A^c\notin\mathcal{T}\Rightarrow$ can not be a discrete topology on $'X'$ which we conclude that $'X'$ is infinite which is a contradiction because $X$ is finite.

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