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Since E is an interval, [a, b]aE and thus SuT=[a, b]. In particular, c = sup S satisfies a^c^b and hence ceS or ceT. 24(3), d(c, S) = 0. 12. If c$T, then ceS. Moreover, because c is an upper bound of S, (c, b] is a non-empty subset of T. Thus d(c, T) = 0 and hence we arrive again at the conclusion that S and T are contiguous. 9 Theorem Let % and y, be metric spaces and let 5c: Z. If / : S-+y, is continuous on the set 5, then S connected =>/(S) connected. Proo/ Let C and D be non-empty subsets of f(S) such that CuD=/(S).

17 Theorem A curve in a metric space % is connected. 9. 18 Path wise connected sets A pathwise connected set is a set with the property that each pair of points in the set can be joined by a curve which lies entirely in the set.

3) Suppose that A and B are sets in a metric space % such that AKJB= %. If A and B are separated, prove that each set is both open and closed. What does this imply about A and B in the metric space 1R"? 1 CONTINUITY Introduction A child takes a lump of modelling clay and by moulding it and pressing it, without ripping or tearing, shapes it into the figure of a man. We say that the original shape is transformed 'continuously' into the final shape. What is it that distinguishes a continuous transformation from any other?

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Commutative algebra [Lecture notes] by Jason P. Bell


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