Regularity is hereditary - Revision history

Vipul: /* Proof */

2012-01-25T00:11:42Z

Proof

← Older revision		Revision as of 00:11, 25 January 2012
Line 70:		Line 70:
	\| 8 \|\| <math>U_1, U_2</math> are the desired open subsets. \|\| \|\| \|\| Steps (5)-(7) \|\| Step-combination, this is exactly what we want to prove.		\| 8 \|\| <math>U_1, U_2</math> are the desired open subsets. \|\| \|\| \|\| Steps (5)-(7) \|\| Step-combination, this is exactly what we want to prove.
	\|}		\|}

			{{tabular proof format}}

	==References==		==References==

2012-01-25T00:05:02Z

Proof

← Older revision		Revision as of 00:05, 25 January 2012
Line 48:		Line 48:

	'''To prove (specific)''': For any point <math>x \in A</math> and any closed subset <math>C</math> of <math>A</math> such that <math>x \notin A</math>, there exist disjoint open subsets <math>U_1, U_2</math> of <math>A</math> such that <math>x \in U_1</math> and <math>C \subseteq U_2</math>.		'''To prove (specific)''': For any point <math>x \in A</math> and any closed subset <math>C</math> of <math>A</math> such that <math>x \notin A</math>, there exist disjoint open subsets <math>U_1, U_2</math> of <math>A</math> such that <math>x \in U_1</math> and <math>C \subseteq U_2</math>.

	~~'''Proof (specific)''':~~

	'''Proof (specific)''':		'''Proof (specific)''':

2012-01-25T00:04:20Z

Proof

← Older revision		Revision as of 00:04, 25 January 2012
Line 47:		Line 47:
	'''Proof''': The T1 property for <math>A</math> follows from Fact (1). It thus suffices to show the separation property for <math>A</math>, i.e., that any point and disjoint closed subset in <math>A</math> can be separated by a continuous function. In other words, we want to prove the following.		'''Proof''': The T1 property for <math>A</math> follows from Fact (1). It thus suffices to show the separation property for <math>A</math>, i.e., that any point and disjoint closed subset in <math>A</math> can be separated by a continuous function. In other words, we want to prove the following.

	'''To prove (specific)''': For any point <math>x \in A</math> and any closed subset <math>C</math> of <math>A</math> such that <math>x \notin A</math>, there exist disjoint open subsets <math>U_1, U_2</math> of <math>A</math> such that <math>x \in U_1<math> and <math>C \subseteq U_2</math>.		'''To prove (specific)''': For any point <math>x \in A</math> and any closed subset <math>C</math> of <math>A</math> such that <math>x \notin A</math>, there exist disjoint open subsets <math>U_1, U_2</math> of <math>A</math> such that <math>x \in U_1</math> and <math>C \subseteq U_2</math>.

	'''Proof (specific)''':		'''Proof (specific)''':

2012-01-25T00:03:59Z

Definitions used

← Older revision		Revision as of 00:03, 25 January 2012
Line 10:		Line 10:

	{{fillin}}		{{fillin}}

			==Related facts==

			===Similar facts===

			{\| class="sortable" border="1"
			! Property !! Proof that is is subspace-hereditary
			\|-
			\| [[Hausdorff space]] \|\| [[Hausdorffness is hereditary]]
			\|-
			\| [[completely regular space]] \|\| [[complete regularity is hereditary]]
			\|-
			\| [[Urysohn space]] \|\| [[Urysohn is hereditary]]
			\|-
			\| [[T1 space]] \|\| [[T1 is hereditary]]
			\|}

			Other similar facts:

			* [[Normality is weakly hereditary]]
			* [[Compactness is weakly hereditary]]

			===Opposite facts===

			* [[Normality is not hereditary]]

	==Facts used==		==Facts used==

2012-01-25T00:02:34Z

Proof

@@ Line 10: / Line 10: @@
 {{fillin}}
 ==Proof==
-===Proof outline===
+'''To prove (specific)''': For any point <math>x \in A</math> and any closed subset <math>C</math> of <math>A</math> such that <math>x \notin A</math>, there exist disjoint open subsets <math>U_1, U_2</math> of <math>A</math> such that <math>x \in U_1<math> and <math>C \subseteq U_2</math>.
-Any subspace of a [[T1 space]] is T1, so we only need to check separation of points and closed sets. We do this as follows:
+'''Proof (specific)''':
-* Pick a point, and a closed set not containing it, in the subspace (the set is closed relative to the subspace)
+'''Proof (specific)''':
-Note that this proof does ''not'' work for normality because we would need to enlarge both [[closed subset]]s, and the process of enlarging might lead to intersection.
+{| class="sortable" border="1"
 ==References==

2012-01-24T21:40:46Z

← Older revision		Revision as of 21:40, 24 January 2012
Line 1:		Line 1:
	{{topospace metaproperty satisfaction}}		{{topospace metaproperty satisfaction\|
			property = regular space\|
	~~{{basic fact~~}}		metaproperty = subspace-hereditary property of topological spaces}}

	==Statement==		==Statement==

	~~===Property-theoretic statement===~~

	~~The [[property of topological spaces]] of being a [[regular space]] is a [[hereditary property of topological spaces]].~~

	~~===Verbal statement===~~

	Any subset of a [[regular space]] is regular under the [[subspace topology]].		Any subset of a [[regular space]] is regular under the [[subspace topology]].
Line 15:		Line 9:
	==Definitions used==		==Definitions used==

	~~===Regular space===~~		{{fillin}}

	{{~~further\|[[Regular space]]~~}}

	~~===Subspace topology===~~

	~~{{further\|[[Subspace topology]]}}~~

	==Proof==		==Proof==

2008-07-20T19:13:17Z

References

← Older revision		Revision as of 19:13, 20 July 2008
Line 39:		Line 39:

	===Textbook references===		===Textbook references===
	* {{booklink\|Munkres}}, Page 196 (Theorem 31.2(b)), Chapter 4, Section 31		* {{booklink-proved\|Munkres}}, Page 196, Theorem 31.2(b), Chapter 4, Section 31

2008-07-20T19:12:55Z

References

← Older revision		Revision as of 19:12, 20 July 2008
Line 39:		Line 39:

	===Textbook references===		===Textbook references===
	* {{booklink\|Munkres}}, Page 196 (Theorem 31.2(b))		* {{booklink\|Munkres}}, Page 196 (Theorem 31.2(b)), Chapter 4, Section 31

2008-05-11T19:57:53Z

2 revisions

← Older revision	Revision as of 19:57, 11 May 2008
(No difference)

2008-04-21T20:23:13Z

← Older revision		Revision as of 20:23, 21 April 2008
Line 35:		Line 35:

	Note that this proof does ''not'' work for normality because we would need to enlarge both [[closed subset]]s, and the process of enlarging might lead to intersection.		Note that this proof does ''not'' work for normality because we would need to enlarge both [[closed subset]]s, and the process of enlarging might lead to intersection.

			==References==

			===Textbook references===
			* {{booklink\|Munkres}}, Page 196 (Theorem 31.2(b))