Basis of topology

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• Neighborhoods and topologies

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.