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Metric Topology http://cis.k.hosei.ac.jp/~yukita/
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2 Neighborhood of a point x in E 1 ・ x-rx-r x x+rx+r N E1E1
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3 Any subset containing a neighborhood is another neighborhood. ・ x-rx-r x x+rx+r N E1E1 N1N1
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4 Accumulation Points
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5 ・ x-r x x+r M m a b ・ x-r x x+r M m a b ・ x-r x x+r M m a b ・ x-r x x+r M m a b The open interval ( a, b ) accumulates at each a < x < b. whatever is the case
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6 ・ x-r x x+r M m a b ・ x-r x x+r M m a b ・ x-r x x+r M m a b ・ x-r x x+r M m a b The closed interval [ a, b ] accumulates at each a b x b b. whatever is the case ・ x-r x=a x+r M m b ・ x-r x=b x+r M m a
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7 Derived Set
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8 Limits of Sequences
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9 Limits of Sequences (Ex12,p.45)
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10 1.1 Prop. A convergent sequence in E 1 has a unique limit. (()) Suppose we have two limits x and y. We can separate them by some of their neighbors as shown below. xy I J
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11 1.2 Monotonic Limits Theorem
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12 Cauchy sequence
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13 1.3 Convergence Characterization
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14 Accumulation and Convergence
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15 1.4 Limit-Accumulation Properties To be filled in the future.
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16 ・ x ・ x ・ x r r r Open n -ball about x with radius r
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17 ・ x ・ x ・ x r r r Closed n -ball about x with radius r
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18 ・ x ・ x ・ x r r r N N N Neighborhood in E n of a point x
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19 2.1 Neighborhood property
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20 Open set
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21 Closed set
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22 Propositions 1.A subset is open in E n if and only if its complement is closed in E n. 2.Any union of open sets is open. 3.Any intersection of closed sets is closed.
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23 A subset is open in E n if and only if its complement is closed in E n. U F
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24 Any union of open sets is open.
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25 Any intersection of closed sets is closed. The dual of the previous proposition
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26 An open set is a union of open balls.
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27 Metric subspaces
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28 Theorem 3.5 Notice that (a) is a special case of (b).
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29 Proof of Th. 3.5(b)
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30 Closure Omitted
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31 Continuity f A f(A)f(A)x f(x)f(x) This kind of situation violates the condition.
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32 Pinching is continuous. ・ ・
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33 Gluing is continuous
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34 4.1 Continuity Characterization
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