Problem 1. For the following, write down an example. No further justification is needed. Please be clear as to what X and d are. 1. A non-empty complete and bounded metric space (X, d) with exactly a countable number of limit points. 2. A non-empty non-complete and unbounded metric space (X, d) with exactly a countable number of limit points. 3. A metric space (X, d) with a non-empty closed an open set ACX such that A ‡ X. 4. A non-empty set X and a function d: X X X → R that satisfies all the properties of a metric except: d(x, y) = 0 implies x = y. 5. A subset of a metric space that is closed and bounded but not compact.
Problem 1. For the following, write down an example. No further justification is needed. Please be clear as to what X and d are. 1. A non-empty complete and bounded metric space (X, d) with exactly a countable number of limit points. 2. A non-empty non-complete and unbounded metric space (X, d) with exactly a countable number of limit points. 3. A metric space (X, d) with a non-empty closed an open set ACX such that A ‡ X. 4. A non-empty set X and a function d: X X X → R that satisfies all the properties of a metric except: d(x, y) = 0 implies x = y. 5. A subset of a metric space that is closed and bounded but not compact.
Problem 1. For the following, write down an example. No further justification is needed. Please be clear as to what X and d are. 1. A non-empty complete and bounded metric space (X, d) with exactly a countable number of limit points. 2. A non-empty non-complete and unbounded metric space (X, d) with exactly a countable number of limit points. 3. A metric space (X, d) with a non-empty closed an open set ACX such that A ‡ X. 4. A non-empty set X and a function d: X X X → R that satisfies all the properties of a metric except: d(x, y) = 0 implies x = y. 5. A subset of a metric space that is closed and bounded but not compact.
I need help coming up with examples and maybe an explanation. The subject is Real Analysis.
Transcribed Image Text:Problem 1. For the following, write down an example. No further justification is needed.
Please be clear as to what X and d are.
1. A non-empty complete and bounded metric space (X, d) with exactly a countable number
of limit points.
2. A non-empty non-complete and unbounded metric space (X, d) with exactly a countable
number of limit points.
3. A metric space (X, d) with a non-empty closed an open set A CX such that A ‡ X.
4. A non-empty set X and a function d : X × X → R that satisfies all the properties of a
metric except: d(x, y) = 0 implies x = y.
5. A subset of a metric space that is closed and bounded but not compact.
Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.
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