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DEFINITION Let ''I'' be a (possibly infinite) (i.e. the topology with the fewest open sets) for which all the projections ''pi'' are Continuous . The product topology is sometimes called the Tychonoff topology. Explicitly, the product topology on ''X'' can be described as the topology generated by sets of the form ''pi''−1(''U''), where ''i'' in ''I'' and ''U'' is an open subset of ''Xi''. In other words, the sets {''pi''−1(''U'')} form a Subbase for the topology on ''X''. A Subset of ''X'' is open if and only if it is a Union of (possibly infinitely many) Intersections of finitely many sets of the form ''pi''−1(''U''). The ''pi''−1(''U'') are sometimes called Open Cylinder s, and their intersections are Cylinder Set s. We can describe a Basis for the product topology using bases of the constituting spaces ''Xi''. Suppose that for each ''i'' in ''I'' we choose a set ''Yi'' which is either the whole space ''Xi'' or a basis set of that space, in such a way that ''Xi'' = ''Yi'' for all but finitely many ''i'' in ''I''. Let ''B'' be the Cartesian product of the sets ''Yi''. The collection of all sets ''B'' that can be constructed in this fashion is a basis of the product space. In particular, this means that a product of finitely many spaces has a basis given by the products of base elements of the ''Xi''. If the index set is finite (in particular, for a product of two topological spaces) then the product topology admits a simpler description. In this case product of the topologies of each ''Xi'' forms a basis for the topology on ''X''. In general, the product of the topologies of each ''Xi'' forms a basis for what is called the Box Topology on ''X''. In general, the box topology is Finer than the product topology, but for finite products they coincide. EXAMPLES If one starts with the standard topology on the Real Line R and defines a topology on the product of ''n'' copies of R in this fashion, one obtains the ordinary Euclidean Topology on R''n''. The Cantor Set is Homeomorphic to the product of Countably Many copies of the Discrete Space {0,1} and the space of Irrational Number s is homeomorphic to the product of countably many copies of the Natural Number s, where again each copy carries the discrete topology. PROPERTIES The product space ''X'', together with the canonical projections, can be characterized by the following : This shows that the product space is a ''fi'' = ''pi'' o ''f'' is continuous for all ''i'' in ''I''. In many cases it is often easier to check that the component functions ''fi'' are continuous. Checking whether a map ''g'' : ''X''→ ''Z'' is continuous is usually more difficult; one tries to use the fact that the ''pi'' are continuous in some way. In addition to being continuous, the canonical projections ''pi'' : ''X'' → ''Xi'' are of the product space whose projections down to all the ''Xi'' are open, then ''W'' need not be open in ''X''. (Consider for instance ''W'' = R2 \ (0,1)2.) The canonical projections are not generally Closed Map s. The product topology is also called the ''topology of pointwise convergence'' because of the following fact: a Sequence (or Net ) in ''X'' converges if and only if all its projections to the spaces ''X''''i'' converge. In particular, if one considers the space ''X'' = R''I'' of all Real valued Function s on ''I'', convergence in the product topology is the same as pointwise convergence of functions. An important theorem about the product topology is is compact. This is easy to show for finite products, while the general statement is equivalent to the Axiom Of Choice . RELATION TO OTHER TOPOLOGICAL NOTIONS
A map that "locally looks like" a canonical projection ''F'' × ''U'' → ''U'' is called a Fiber Bundle . SEE ALSO |
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