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Notation and basic conventions. Throughout let X and Y denote topological spaces, and for a subset A of a topological space denote its closure by A. 1. State precise technical definitions of four of the following . closure. . limit point, . compact, and . product topology (on a not necessarily countable product of topological spaces). 2. Prove ONE of the following two (you will earn credit for only one). . Every closed subset of every compact space is compact. . The continuous image of every path connected space is path connected. 3. In which of the product, uniform, and box topologies do the sequences (a,)-1, (b.), C RN converge? Justify your answers. = (1, 1, 1, 1, 1, 1, 1, ...) = (1. . ..) 02 (1, 1. 2, 4, 8, 16, 32, ...) by = (1, 38. ... ) 0s = (1. 1, 1, 3, 9, 27. 81, . ..) 4 - (1, 1, 1, 1, 4, 16, 64, ...) by - (1, 161 871 261 68' 126 ...) * = max(0.A-] bnk = 4. and 5. Prove TWO of the following (you will earn credit for only two). . If f : X - Y and for every subset A C X, /(A) C /(A), then f is continuous. . The subsets of bounded and unbounded sequences form a separation of R" in the uniform topology. . If A C X is connected, and A C B C A then B is connected. . If {In : a C A) is a collection of Hausdorff spaces, then the product IIneA Xo is Hausdorff in both the product and box topologies. Bonus. Demonstrate that the subset [-1, 1] C Re with the lower limit topology is neither compact nor sequentially compact