My topology text defines the "neighborhood" of a point as follows:Let p be a point in a topological space X .A subset N of X is a neighborhood of p iff N is a superset of an open set G containing p. Now I am stuck on the following problem:Let X be a cofinite topological space .Show that every neighborhood of a point p (element of x) is an open set .Now since in cofinite topology the closed sets are finite , so this should be true by definition , that is aclosed set could not contain an open set ...I just cannot see what we have to prove here ........
The definition is correct and also the statement that every neighborhood of a point in a space endowed with the cofinite topology is open.
Suppose $N$ is a neighborhood of $p$ and that $G$ is an open set such that $p\in G$ and $G\subseteq N$.
Then $X\setminus G\supseteq X\setminus N$, so $X\setminus N$ is finite. Hence $N$ is open.