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Consider R with the standard topology as well as R ‘: the real numbers with the lower limit topology, whose basis consists of the intervals [a,b). A class B of open sets is a base for the topology of X if each open set of X is the union of some of the members of B. Syn. 2. Product, Box, and Uniform Topologies 18 If B is a basis for the topology of X and C is a basis for the topology of Y, then the collection D = {B × C | B ∈ B and C ∈ C} is a basis for the topology of X ×Y. Examples: [of bases] (i) Open intervals of the form pa;bqare a basis for the standard topology on R. (ii) Interior of circle are a basis for the standard topology in R2. Basis for a Topology 4 4. Topology Generated by a Basis 4 4.1. In nitude of Prime Numbers 6 5. careful, we should really say that we are using the standard absolute value metric on R and the corresponding metric topology — the usual topology to use for R.) An example that is perhaps more satisfying is fz= x+iy2C : 0 x;y<1g. The topology generated by this basis is the topology in which the open sets are precisely the unions of basis sets. Basis. The standard topology on R2 is the product topology on R×R where we have the standard topology on R. In symbols: if is a set, a collection of subsets of is said to form a basis for a topology on if the following two conditions are satisfied: For all , … (b) Determine all continuous maps f : R ‘ → R. 3. It is again neither open Any such set can be decomposed as the union S a

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