CRN 11982: HW 6 - Revision history

HelmutKnaust at 02:18, 29 October 2014

2014-10-29T02:18:56Z

HelmutKnaust at 02:18, 29 October 2014

2014-10-29T02:18:21Z

HelmutKnaust at 02:16, 29 October 2014

2014-10-29T02:16:02Z

HelmutKnaust at 01:24, 29 October 2014

2014-10-29T01:24:41Z

HelmutKnaust at 01:23, 29 October 2014

2014-10-29T01:23:59Z

HelmutKnaust at 00:40, 29 October 2014

2014-10-29T00:40:53Z

HelmutKnaust at 00:40, 29 October 2014

2014-10-29T00:40:13Z

HelmutKnaust: Created page with "'''Problem 26.''' Exercise 3.3.3 '''Problem 27.''' Exercise 3.3.7 (b,c,e) '''Problem 28.''' Exercise 3.3.9 (b,f) '''Problem 29.''' Exercise 4.2.9 '''Problem 30.''' A set $..."

2014-10-29T00:32:55Z

Created page with "'''Problem 26.''' Exercise 3.3.3 '''Problem 27.''' Exercise 3.3.7 (b,c,e) '''Problem 28.''' Exercise 3.3.9 (b,f) '''Problem 29.''' Exercise 4.2.9 '''Problem 30.''' A set $..."

New page

'''Problem 26.''' Exercise 3.3.3

'''Problem 27.''' Exercise 3.3.7 (b,c,e)

'''Problem 28.''' Exercise 3.3.9 (b,f)

'''Problem 29.''' Exercise 4.2.9

'''Problem 30.''' A set $X$ is called ''LP-compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $X$.

Show that a set is LP-compact if and only if it is closed and bounded.

Hint: Show first that every bounded infinite set has a limit point.

@@ Line 9: / Line 9: @@
 Show that a set is LP-compact if and only if it is closed and bounded.
-Hint: Recall that every bounded infinite set has a limit point.
+* Hint: Recall that every bounded infinite set has a limit point.
 '''Problem 30.''' Show that every non-empty open set is the countable union of pairwise disjoint open intervals: Let $O$ be a non-empty open set of real numbers. Then there are open intervals $I_n$, $n\in \mathbb{N}$, such that
 * $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and
 * for all $m,n\in\mathbb{N}$:   $I_m\cap I_n=\emptyset$ or $I_m=I_n$.

← Older revision		Revision as of 02:18, 29 October 2014
Line 11:		Line 11:
	Hint: Recall that every bounded infinite set has a limit point.		Hint: Recall that every bounded infinite set has a limit point.

−	'''Problem 30.''' Show that every non-empty open set is the countable union of pairwise disjoint open intervals: Let $O$ be a non-empty open set of real numbers. Then there are open intervals $I_n$, $n\in \mathbb{N}$, such that ~~(1)~~ $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and ~~(2)~~ for all $m,n\in\mathbb{N}$: $I_m\cap I_n=\emptyset$ or $I_m=I_n$.	+	'''Problem 30.''' Show that every non-empty open set is the countable union of pairwise disjoint open intervals: Let $O$ be a non-empty open set of real numbers. Then there are open intervals $I_n$, $n\in \mathbb{N}$, such that
		+	* $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and
		+	* for all $m,n\in\mathbb{N}$: $I_m\cap I_n=\emptyset$ or $I_m=I_n$.

← Older revision		Revision as of 02:16, 29 October 2014
Line 11:		Line 11:
	Hint: Recall that every bounded infinite set has a limit point.		Hint: Recall that every bounded infinite set has a limit point.

−	'''Problem 30.''' Show the ~~following~~: Let $O$ be a non-empty open set of real numbers. Then there are open intervals $I_n$, $n\in \mathbb{N}$, such that (1) $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and (2) $I_m\cap I_n=\emptyset$ ~~for all~~ $~~m\neq n~~$.	+	'''Problem 30.''' Show that every non-empty open set is the countable union of pairwise disjoint open intervals: Let $O$ be a non-empty open set of real numbers. Then there are open intervals $I_n$, $n\in \mathbb{N}$, such that (1) $\displaystyle O=\bigcup_{n\in\mathbb{N}} I_n$, and (2) for all $m,n\in\mathbb{N}$: $I_m\cap I_n=\emptyset$ or $I_m=I_n$.

@@ Line 5: / Line 5: @@
 '''Problem 28.''' Exercise 3.3.10
-'''Problem 29.''' A set $X$ is called ''LP-compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $X$.
+'''Problem 29.''' A set $X$ is called ''LP-compact'', if every infinite subset of $X$ has a limit point belonging to $X$.
 Show that a set is LP-compact if and only if it is closed and bounded.

@@ Line 1: / Line 1: @@
 '''Problem 26.''' Exercise 3.3.1
-'''Problem 27.''' Exercise 3.3.7 (b,c,e)
+'''Problem 27.''' Exercise 3.3.9 (b,e)
-'''Problem 28.''' Exercise 3.3.9 (b,e)
+'''Problem 28.''' Exercise 3.3.10
-'''Problem 29.''' Exercise 3.3.10
+'''Problem 29.''' A set $X$ is called ''LP-compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $X$.
-−
-'''Problem 30.''' A set $X$ is called ''LP-compact'', if every infinite subset $A$ of $X$ has a limit point belonging to $X$.
 Show that a set is LP-compact if and only if it is closed and bounded.
 Hint: Recall that every bounded infinite set has a limit point.

← Older revision		Revision as of 00:40, 29 October 2014
Line 11:		Line 11:
	Show that a set is LP-compact if and only if it is closed and bounded.		Show that a set is LP-compact if and only if it is closed and bounded.

−	Hint: ~~Show first~~ that every bounded infinite set has a limit point.	+	Hint: Recall that every bounded infinite set has a limit point.