Vipul: /* Proof */

2009-10-26T20:22:20Z

Proof

← Older revision		Revision as of 20:22, 26 October 2009
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	'''To prove''': If <math>\{ x_n \}_{n \in \mathbb{N}}</math> is a sequence with limit <math>x</math> and limit <math>y</math>, then <math>x = y</math>.		'''To prove''': If <math>\{ x_n \}_{n \in \mathbb{N}}</math> is a sequence with limit <math>x</math> and limit <math>y</math>, then <math>x = y</math>.

	'''Proof''': Clearly, <math>x_n</math> cannot be eventually constant at both <math>x</math> and <math>y</math>. Let us say that it is not eventually constant at <math>y</math>. Throw out all the <math>x_n</math>s that are ~~not~~ equal to <math>y</math>. This new sequence of <math>x_n</math>s converges to both <math>x</math> and <math>y</math>. Let <math>A</math> be the union of <math>\{ x \}</math> and the <math>x_n</math>s in this new sequence.		'''Proof''': Clearly, <math>x_n</math> cannot be eventually constant at both <math>x</math> and <math>y</math>. Let us say that it is not eventually constant at <math>y</math>. Throw out all the <math>x_n</math>s that are equal to <math>y</math>. This new sequence of <math>x_n</math>s converges to both <math>x</math> and <math>y</math>. Let <math>A</math> be the union of <math>\{ x \}</math> and the <math>x_n</math>s in this new sequence.

	# <math>A</math> is compact: Any open cover of <math>A</math> contains one open set containing <math>A</math>, so it contains all but finitely many of the <math>x_n</math>s. Thus, that along with finitely many other open subsets covers <math>A</math>. Thus, every open cover has a finite subcover.		# <math>A</math> is compact: Any open cover of <math>A</math> contains one open set containing <math>A</math>, so it contains all but finitely many of the <math>x_n</math>s. Thus, that along with finitely many other open subsets covers <math>A</math>. Thus, every open cover has a finite subcover.

Vipul: Created page with '{{topospace property implication| stronger = KC-space| weaker = US-space}} ==Statement== Any KC-space is a US-space. ==Definitions used== ===KC-space=== {{further|[[…'

2009-10-26T20:08:03Z

Created page with '{{topospace property implication| stronger = KC-space| weaker = US-space}} ==Statement== Any KC-space is a US-space. ==Definitions used== ===KC-space=== {{further|[[…'

New page

{{topospace property implication|
stronger = KC-space|
weaker = US-space}}

==Statement==

Any [[KC-space]] is a [[US-space]].

==Definitions used==

===KC-space===

{{further|[[KC-space]]}}

A topological space is termed a KC-space if every compact subset of it is closed.

===US-space===

{{further|[[US-space]]}}

A topological space is termed a US-space if every convergent sequence has a unique limit.

==Proof==

'''Given''': A KC-space <math>X</math>.

'''To prove''': If <math>\{ x_n \}_{n \in \mathbb{N}}</math> is a sequence with limit <math>x</math> and limit <math>y</math>, then <math>x = y</math>.

'''Proof''': Clearly, <math>x_n</math> cannot be eventually constant at both <math>x</math> and <math>y</math>. Let us say that it is not eventually constant at <math>y</math>. Throw out all the <math>x_n</math>s that are not equal to <math>y</math>. This new sequence of <math>x_n</math>s converges to both <math>x</math> and <math>y</math>. Let <math>A</math> be the union of <math>\{ x \}</math> and the <math>x_n</math>s in this new sequence.

# <math>A</math> is compact: Any open cover of <math>A</math> contains one open set containing <math>A</math>, so it contains all but finitely many of the <math>x_n</math>s. Thus, that along with finitely many other open subsets covers <math>A</math>. Thus, every open cover has a finite subcover.
# <math>A</math> is closed: This follows from the previous step, and the assumption that <math>X</math> is a KC-space.
# There is an open subset containing <math>y</math> that does not contain any of the <math>x_n</math>s: This follows from the fact that <math>A</math> is closed, and <math>y \notin A</math>.

The last step contradicts our assumption that the <math>x_n</math>s converge to <math>y</math>.

KC implies US - Revision history

Vipul: /* Proof */

Vipul: Created page with '{{topospace property implication| stronger = KC-space| weaker = US-space}} ==Statement== Any KC-space is a US-space. ==Definitions used== ===KC-space=== {{further|[[…'