P. 156 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 15501-15600 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	xblpnfps 15501	The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet $/\$ $/\$

Theorem	xblpnf 15502	The infinity ball in an extended metric is the set of all points that are a finite distance from the center. (Contributed by Mario Carneiro, 23-Aug-2015.)
$/\$ $/\$

Theorem	blpnf 15503	The infinity ball in a standard metric is just the whole space. (Contributed by Mario Carneiro, 23-Aug-2015.)
$/\$

Theorem	bldisj 15504	Two balls are disjoint if the center-to-center distance is more than the sum of the radii. (Contributed by Mario Carneiro, 30-Dec-2013.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	blgt0 15505	A nonempty ball implies that the radius is positive. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
$/\$ $/\$ $/\$

Theorem	bl2in 15506	Two balls are disjoint if they don't overlap. (Contributed by NM, 11-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
$/\$ $/\$ $/\$ $/\$

Theorem	xblss2ps 15507	One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 15510 for extended metrics, we have to assume the balls are a finite distance apart, or else will not even be in the infinity ball around . (Contributed by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theorem	xblss2 15508	One ball is contained in another if the center-to-center distance is less than the difference of the radii. In this version of blss2 15510 for extended metrics, we have to assume the balls are a finite distance apart, or else will not even be in the infinity ball around . (Contributed by Mario Carneiro, 23-Aug-2015.)


Theorem	blss2ps 15509	One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet $/\$ $/\$ $/\$ $/\$ $/\$

Theorem	blss2 15510	One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	blhalf 15511	A ball of radius is contained in a ball of radius centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)
$/\$ $/\$ $/\$

Theorem	blfps 15512	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theorem	blf 15513	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)


Theorem	blrnps 15514*	Membership in the range of the ball function. Note that is the collection of all balls for metric . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theorem	blrn 15515*	Membership in the range of the ball function. Note that is the collection of all balls for metric . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)


Theorem	xblcntrps 15516	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet $/\$ $/\$ $/\$

Theorem	xblcntr 15517	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$ $/\$

Theorem	blcntrps 15518	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet $/\$ $/\$

Theorem	blcntr 15519	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	xblm 15520*	A ball is inhabited iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
$/\$ $/\$

Theorem	bln0 15521	A ball is not empty. It is also inhabited, as seen at blcntr 15519. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	blelrnps 15522	A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet $/\$ $/\$

Theorem	blelrn 15523	A ball belongs to the set of balls of a metric space. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	blssm 15524	A ball is a subset of the base set of a metric space. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	unirnblps 15525	The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet

Theorem	unirnbl 15526	The union of the set of balls of a metric space is its base set. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)


Theorem	blininf 15527	The intersection of two balls with the same center is the smaller of them. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$ $/\$ inf

Theorem	ssblps 15528	The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet $/\$ $/\$ $/\$ $/\$

Theorem	ssbl 15529	The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
$/\$ $/\$ $/\$ $/\$

Theorem	blssps 15530*	Any point in a ball can be centered in another ball that is a subset of . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet $/\$ $/\$

Theorem	blss 15531*	Any point in a ball can be centered in another ball that is a subset of . (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
$/\$ $/\$

Theorem	blssexps 15532*	Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
PsMet $/\$ $/\$

Theorem	blssex 15533*	Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	ssblex 15534*	A nested ball exists whose radius is less than any desired amount. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	blin2 15535*	Given any two balls and a point in their intersection, there is a ball contained in the intersection with the given center point. (Contributed by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$ $/\$

Theorem	blbas 15536	The balls of a metric space form a basis for a topology. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 15-Jan-2014.)


Theorem	blres 15537	A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
$/\$ $/\$

Theorem	xmeterval 15538	Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
$/\$ $/\$

Theorem	xmeter 15539	The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)


Theorem	xmetec 15540	The equivalence classes under the finite separation equivalence relation are infinity balls. (Contributed by Mario Carneiro, 24-Aug-2015.)
$/\$

Theorem	blssec 15541	A ball centered at is contained in the set of points finitely separated from . This is just an application of ssbl 15529 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
$/\$ $/\$

Theorem	blpnfctr 15542	The infinity ball in an extended metric acts like an ultrametric ball in that every point in the ball is also its center. (Contributed by Mario Carneiro, 21-Aug-2015.)
$/\$ $/\$

Theorem	xmetresbl 15543	An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 15540, this shows that any extended metric space can be "factored" into the disjoint union of proper metric spaces, with points in the same region measured by that region's metric, and points in different regions being distance from each other. (Contributed by Mario Carneiro, 23-Aug-2015.)
$/\$ $/\$

9.2.4 Open sets of a metric space

Theorem	mopnrel 15544	The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)


Theorem	mopnval 15545	An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object is the family of all open sets in the metric space determined by the metric . By mopntop 15547, the open sets of a metric space form a topology , whose base set is by mopnuni 15548. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)


Theorem	mopntopon 15546	The set of open sets of a metric space is a topology on . Remark in [Kreyszig] p. 19. This theorem connects the two concepts and makes available the theorems for topologies for use with metric spaces. (Contributed by Mario Carneiro, 24-Aug-2015.)
TopOn

Theorem	mopntop 15547	The set of open sets of a metric space is a topology. (Contributed by NM, 28-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)


Theorem	mopnuni 15548	The union of all open sets in a metric space is its underlying set. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)


Theorem	elmopn 15549*	The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	mopnfss 15550	The family of open sets of a metric space is a collection of subsets of the base set. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)


Theorem	mopnm 15551	The base set of a metric space is open. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)


Theorem	elmopn2 15552*	A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$

Theorem	mopnss 15553	An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)
$/\$

Theorem	isxms 15554	Express the predicate " is an extended metric space" with underlying set and distance function . (Contributed by Mario Carneiro, 2-Sep-2015.)
$/\$

Theorem	isxms2 15555	Express the predicate " is an extended metric space" with underlying set and distance function . (Contributed by Mario Carneiro, 2-Sep-2015.)
$/\$

Theorem	isms 15556	Express the predicate " is a metric space" with underlying set and distance function . (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
$/\$

Theorem	isms2 15557	Express the predicate " is a metric space" with underlying set and distance function . (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
$/\$

Theorem	xmstopn 15558	The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)


Theorem	mstopn 15559	The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)


Theorem	xmstps 15560	An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)


Theorem	msxms 15561	A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)


Theorem	mstps 15562	A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)


Theorem	xmsxmet 15563	The distance function, suitably truncated, is an extended metric on . (Contributed by Mario Carneiro, 2-Sep-2015.)


Theorem	msmet 15564	The distance function, suitably truncated, is a metric on . (Contributed by Mario Carneiro, 12-Nov-2013.)


Theorem	msf 15565	The distance function of a metric space is a function into the real numbers. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)


Theorem	xmsxmet2 15566	The distance function, suitably truncated, is an extended metric on . (Contributed by Mario Carneiro, 2-Oct-2015.)


Theorem	msmet2 15567	The distance function, suitably truncated, is a metric on . (Contributed by Mario Carneiro, 2-Oct-2015.)


Theorem	mscl 15568	Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$

Theorem	xmscl 15569	Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$

Theorem	xmsge0 15570	The distance function in an extended metric space is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)
$/\$ $/\$

Theorem	xmseq0 15571	The distance between two points in an extended metric space is zero iff the two points are identical. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$

Theorem	xmssym 15572	The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$

Theorem	xmstri2 15573	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	mstri2 15574	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	xmstri 15575	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	mstri 15576	Triangle inequality for the distance function of a metric space. Definition 14-1.1(d) of [Gleason] p. 223. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	xmstri3 15577	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	mstri3 15578	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	msrtri 15579	Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
$/\$ $/\$ $/\$

Theorem	xmspropd 15580	Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)


Theorem	mspropd 15581	Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)


Theorem	setsmsbasg 15582	The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theorem	setsmsdsg 15583	The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
sSet TopSet

Theorem	setsmstsetg 15584	The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)
sSet TopSet TopSet

Theorem	mopni 15585*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$ $/\$

Theorem	mopni2 15586*	An open set of a metric space includes a ball around each of its points. (Contributed by NM, 2-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	mopni3 15587*	An open set of a metric space includes an arbitrarily small ball around each of its points. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$ $/\$ $/\$

Theorem	blssopn 15588	The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)


Theorem	unimopn 15589	The union of a collection of open sets of a metric space is open. Theorem T2 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
$/\$

Theorem	mopnin 15590	The intersection of two open sets of a metric space is open. (Contributed by NM, 4-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
$/\$ $/\$

Theorem	mopn0 15591	The empty set is an open set of a metric space. Part of Theorem T1 of [Kreyszig] p. 19. (Contributed by NM, 4-Sep-2006.)


Theorem	rnblopn 15592	A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)
$/\$

Theorem	blopn 15593	A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	neibl 15594*	The neighborhoods around a point of a metric space are those subsets containing a ball around . Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
$/\$ $/\$

Theorem	blnei 15595	A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
$/\$ $/\$

Theorem	blsscls2 15596*	A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
$/\$ $/\$ $/\$ $/\$

Theorem	metss 15597*	Two ways of saying that metric generates a finer topology than metric . (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
$/\$

Theorem	metequiv 15598*	Two ways of saying that two metrics generate the same topology. Two metrics satisfying the right-hand side are said to be (topologically) equivalent. (Contributed by Jeff Hankins, 21-Jun-2009.) (Revised by Mario Carneiro, 12-Nov-2013.)
$/\$ $/\$

Theorem	metequiv2 15599*	If there is a sequence of radii approaching zero for which the balls of both metrics coincide, then the generated topologies are equivalent. (Contributed by Mario Carneiro, 26-Aug-2015.)
$/\$ $/\$

Theorem	metss2lem 15600*	Lemma for metss2 15601. (Contributed by Mario Carneiro, 14-Sep-2015.)
$/\$ $/\$ $/\$ $/\$

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