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Basis of topology
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Suppose a set $X$ is given. Any collection $\gamma$ of subsets of $X$ is called a basis of topology in $X$ if it satisfies the following conditions:
(B1) $\gamma$ covers the whole $X: U \gamma = X$.
(B2) For any $U, V \in \gamma$ and any point $x \in U \cap V$, there is a $W \in \gamma$ such that $x \in W < U \cap V$.
The elements of $\gamma$ are called neighborhoods and if $x \in W$ we say that $W$ is a neighborhood of $x$.
For example, the collection of all "open" balls in $X = {\bf R}^n$:
$\gamma_b = \{B(a, \delta): a \in {\bf R}^n, \delta > 0 \}$, where $B(p,d)= \{u: |u-p| < d \}$
is a basis. Rectangles, ellipses, etc also work.
For more see Neighborhoods and topologies.