Posted by J. Holmer on September 25, 2007, 8:15 pm, in reply to "Bases" { (n, n+2) | n integer } is an open cover of R but not a base for the usual topology. In order to be a base, we need to have that for each open set U and each x_0\in U, there exists a set V in the base such that x_0 \in V\subset U. So in our example, if we took U=(-1/2, 1/2), and x_0=0, there would be no set of the form (n,n+2) contained in (-1/2, 1/2). So a base needs to include a sufficient number of "small" open sets in all regions of the space, whereas an open cover does not. (2) The concept of base in a topological space and a basis of a vector space are analogous concepts, but there is an important difference. In a topological space, any open set can be written as a union of base elements. In a vector space, any element of the space can be represented as a linear combination of elements from the basis. This is the similarity: the topology (that is, the open sets) can be reconstructed from the base in a natural way considering the available structure (via unions) and the elements of a vector space can be reconstructed from a basis by using the available algebraic structure (multiplication by scalars and addition). But an important difference is that the representation of an open set as a union of base elements is (typically) not unique. The notion of base as a subcollection from which general objects can be represented (or "reconstructed" ) appears in several places in math. Another example is that of a transcendence basis for a field extension in algebra.
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(1) Let (X,T) be a topological space. If B is a base for (X,T), then it is an open cover of X. But a given open cover { U_i } need not be a base. There are many examples, but this one is illustrative: Take X=R (the real line) and T= the usual topology. Then the set of all intervals of the form 143
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