Algebra moderna by Frank Ayres Jr.

March 9, 2017 | Mathematics | By admin | 0 Comments

By Frank Ayres Jr.

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It is obvious that if we set Po = pi (2 jl) where p is as above, then Po is a bilinear mapping of 2 j{ into C x :F. 2. Indeed, if

If U is a connected open subset of IRn, then U is arcwise connected. PROOF. We may assume U to be nonempty. Let Xo be a point of U, and denote by V the set of those points x in U such that there exists an arc in U joining Xo to x. If x E V and if W is an open cell in IR n containing x and contained in U (Ex. A), then x can clearly be joined to every point of W by an arc in W Hence W c V, and it follows that V is an open set. Suppose now that Yo E Un av, and let W1 be an open cell containing Yo and contained in U.

If V is merely allowed to run over a base [subbase] for the topology on Y, then the inverse images l-l(V) provide a base [subbase] for the inversely induced topology on X. Example L. Let X be a topological space, and let A be a subset of X. If we take for I the inclusion mapping of A into X (f(x) = x for all x in A), then the topology inversely induced on A by I is the relative topology on A. 42 3 General topology A more sophisticated version of the above construction goes as follows. Definition.

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