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41. Prove: If X has a nite number of components, then each component is both open and closed. 42. Prove: If E is a non-empty connected subset of X which is both open and closed, then E is a component. i3. Prove: Let E be a component of Y and let 3' : X Y be continuous. Then f'1[E] is a union of components of X. \\ 44. Prove: Let X be a compact space. If the components of X are open, then there are only a nite number of them: ARCWISE CONNECTED SETS 45. Show that an indiscrete space is arcwise connected. 46. Prove: The arcwise connected components of X form a partition of X. 47. Prov'e: EVery component of X is partitioned by arcwiso connected components. MISCELLANEOUS PROBLEMS 48. Show that an indiscrete space is simply connected. 49. Show that a totally disconnected space is Hausdorff. 50. Prove: Let G be an open subset of a locally connected space X. Then G is locally connected. El. Let A : {c.,b} he discrete and-let i = {0.1}. Show thst the product space X = H {A}: A} = A, i E if} is not locally connected. Hence locally connectedness is not product invariant. 52. Show that \"simply connected\" is a topological property. 53. Prove: Let X be locally connected. Then 'X is connected if and only if there exists a simple chain of connectedqets joining any pair of points in X