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Proposition: Properties of the Set of All Neighborhoods of a Point

Let $(X,\mathcal O)$ be a topological space, $A\in X$ be a point and $\mathcal N(A)$ be the set of all neighborhoods of $A.$ Then $\mathcal N(A)$ has the following properties:

1. $\mathcal N(A)\neq \emptyset$ (it is not empty).
2. $\emptyset\not\in\mathcal N(A)$ (the empty set is not a neighbourhood of $A$).
3. $U,W\in\mathcal N(A)\Rightarrow U\cap W\in\mathcal N(A)$ (the intersection any two neighborhoods of $A$ is also a neighborhood of $A$).
4. $U,W\in\mathcal N(A)\Rightarrow U\cap W\in\mathcal N(A)$ (the intersection any two neighborhoods of $A$ is also a neighborhood of $A$).

Table of Contents

Proofs: 1

Mentioned in:

Axioms: 1

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References

Bibliography

1. Grotemeyer, K.P.: "Topologie", B.I.-Wissenschaftsverlag, 1969