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Definition: If X is a set, a basis for a topology on X is a collection B of subsets of X (called basis elements) such that: (1) For each x in X, there is at least one basis element b containing x. (2) If x belongs to the intersection of two basis elements b_1 and b_2, then there is a basis element b_3 containing x such that b_3 is a subset of (b_1 intersection b_2).
Source: Topology (second edition) by James R. Munkres
Example: The basis set is often denoted with a script B, and elements of the basis are often denoted as regular B_i, however, due to formatting constraints we substituted the above notation.