By Yuying Q.

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5. 8. Let (X, d) be a metric space, and let E be a subset of X. Show that if E is connected, then the closure E of E is also connected. Is the converse true? 9. Let (X, d) be a metric space. Let us define a relation x "' y on X by declaring x "' y iff there exists a connected subset of X which contains both x andy. , it obeys the reflexive, symmetric, and transitive axioms). , the sets of the form {y E X : y "' x} for some x E X) are all closed and connected. ) These sets are known as the connected components of X.

Then there exists a real number x < z < y such that z ¢ X. Thus the sets (-oo, z) n X and (z, oo) n X will cover X. But these sets are non-empty (because they contain x and y respectively) and are open relative to X, and so, X is disconnected, a contradiction. Now we show that (b) implies (a). Let X be a set obeying the property {b). Suppose for sake of contradiction that X is disconnected. Then there exist disjoint non-empty sets V, W which are open relative to X, such that V U W = X. Since V and W are non-empty, we may choose an x E V and y E W.

19. 14. 20. 15. Let 00 X:= {(an):=o: L lanl < oo} n=O be the space of absolutely convergent sequences. Define the l 1 and metrics on this space by 00 dtl((an):=O• (bn):=o) := L n=O ian- bnl; zoo 400 12. Metric spaces Show that these are both metrics on X, but show that there exist sequences x< 1>, x< 2 >, ... , sequences of sequences) which are convergent with respect to the d1oo metric but not with respect to the d11 metric. Conversely, show that any sequence which converges in the dp metric automatically converges in the d1oo metric.

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A Boundary Value Problem for Hypermon Genic Functions in Clifford Analysis by Yuying Q.


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