Answer to Question #141477 in Differential Geometry | Topology for kurd math

Question

Q/in cofinite topology (X,C), P∈X, Show that N(p) subset and equel to C were N(p) is set of all neignborhoods of p.

Answer 1

Let $(X, C)$ be a topological space, where $C$ is cofinite topology on $X$. Let us show that $N(p)\subseteq C$ where $N(p)$ is set of all neignborhoods of $p\in X$.

By defenition, a subset $U\subseteq X$ is called an open set of topological space $(X, C)$ if $U$ is an element of topology $C$. A neignborhood of a point $p$ is an open set $U$ such that $p\in U$. It follows that each neignborhood of $p$ belongs to $C$. Therefore, $N(p)\subseteq C$.

On the other hand, the set $V=X\setminus\{p\}$ is an open set in cofinite topology $C$ on $X$, and $p\notin V$. Therefore, $N(p)\ne C.$