Question #141477
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.
1
Expert's answer
2020-11-02T17:42:03-0500

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


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


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




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