P. 248 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24701-24800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	xblss2 24701	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 24703 for extended metrics, we have to assume the balls are a finite distance apart, or else 𝑃 will not even be in the infinity ball around 𝑄. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑄 ∈ 𝑋)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑆 ∈ ℝ*)    &   ⊢ (𝜑 → (𝑃𝐷𝑄) ∈ ℝ)    &   ⊢ (𝜑 → (𝑃𝐷𝑄) ≤ (𝑆 +𝑒 -𝑒𝑅))    ⇒   ⊢ (𝜑 → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
Theorem	blss2ps 24702	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.)
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆 − 𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
Theorem	blss2 24703	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.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ (𝑃𝐷𝑄) ≤ (𝑆 − 𝑅))) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑄(ball‘𝐷)𝑆))
Theorem	blhalf 24704	A ball of radius 𝑅 / 2 is contained in a ball of radius 𝑅 centered at any point inside the smaller ball. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Jan-2014.)
⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑌 ∈ 𝑋) ∧ (𝑅 ∈ ℝ ∧ 𝑍 ∈ (𝑌(ball‘𝑀)(𝑅 / 2)))) → (𝑌(ball‘𝑀)(𝑅 / 2)) ⊆ (𝑍(ball‘𝑀)𝑅))
Theorem	blfps 24705	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
Theorem	blf 24706	Mapping of a ball. (Contributed by NM, 7-May-2007.) (Revised by Mario Carneiro, 23-Aug-2015.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → (ball‘𝐷):(𝑋 × ℝ*)⟶𝒫 𝑋)
Theorem	blrnps 24707*	Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))
Theorem	blrn 24708*	Membership in the range of the ball function. Note that ran (ball‘𝐷) is the collection of all balls for metric 𝐷. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ ran (ball‘𝐷) ↔ ∃𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ* 𝐴 = (𝑥(ball‘𝐷)𝑟)))
Theorem	xblcntrps 24709	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.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	xblcntr 24710	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ (𝑅 ∈ ℝ* ∧ 0 < 𝑅)) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	blcntrps 24711	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.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	blcntr 24712	A ball contains its center. (Contributed by NM, 2-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → 𝑃 ∈ (𝑃(ball‘𝐷)𝑅))
Theorem	xbln0 24713	A ball is nonempty iff the radius is positive. (Contributed by Mario Carneiro, 23-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → ((𝑃(ball‘𝐷)𝑅) ≠ ∅ ↔ 0 < 𝑅))
Theorem	bln0 24714	A ball is not empty. (Contributed by NM, 6-Oct-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ≠ ∅)
Theorem	blelrnps 24715	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.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
Theorem	blelrn 24716	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.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ ran (ball‘𝐷))
Theorem	blssm 24717	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.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ⊆ 𝑋)
Theorem	unirnblps 24718	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.)
⊢ (𝐷 ∈ (PsMet‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
Theorem	unirnbl 24719	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.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → ∪ ran (ball‘𝐷) = 𝑋)
Theorem	blin 24720	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.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ*)) → ((𝑃(ball‘𝐷)𝑅) ∩ (𝑃(ball‘𝐷)𝑆)) = (𝑃(ball‘𝐷)if(𝑅 ≤ 𝑆, 𝑅, 𝑆)))
Theorem	ssblps 24721	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.)
⊢ (((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ*) ∧ 𝑅 ≤ 𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))
Theorem	ssbl 24722	The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ* ∧ 𝑆 ∈ ℝ*) ∧ 𝑅 ≤ 𝑆) → (𝑃(ball‘𝐷)𝑅) ⊆ (𝑃(ball‘𝐷)𝑆))
Theorem	blssps 24723*	Any point 𝑃 in a ball 𝐵 can be centered in another ball that is a subset of 𝐵. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)
Theorem	blss 24724*	Any point 𝑃 in a ball 𝐵 can be centered in another ball that is a subset of 𝐵. (Contributed by NM, 31-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷) ∧ 𝑃 ∈ 𝐵) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐵)
Theorem	blssexps 24725*	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.)
⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴))
Theorem	blssex 24726*	Two ways to express the existence of a ball subset. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴) ↔ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝐴))
Theorem	ssblex 24727*	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.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) ∧ (𝑅 ∈ ℝ+ ∧ 𝑆 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ (𝑃(ball‘𝐷)𝑆)))
Theorem	blin2 24728*	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.)
⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝐵 ∩ 𝐶)) ∧ (𝐵 ∈ ran (ball‘𝐷) ∧ 𝐶 ∈ ran (ball‘𝐷))) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ (𝐵 ∩ 𝐶))
Theorem	blbas 24729	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.)
⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ∈ TopBases)
Theorem	blres 24730	A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
⊢ 𝐶 = (𝐷 ↾ (𝑌 × 𝑌))    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋 ∩ 𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑅) = ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌))
Theorem	xmeterval 24731	Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ∼ = (◡𝐷 “ ℝ)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∼ 𝐵 ↔ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ)))
Theorem	xmeter 24732	The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ∼ = (◡𝐷 “ ℝ)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∼ Er 𝑋)
Theorem	xmetec 24733	The equivalence classes under the finite separation equivalence relation are infinity balls. Thus, by erdisj 8759, infinity balls are either identical or disjoint, quite unlike the usual situation with Euclidean balls which admit many kinds of overlap. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ∼ = (◡𝐷 “ ℝ)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → [𝑃] ∼ = (𝑃(ball‘𝐷)+∞))
Theorem	blssec 24734	A ball centered at 𝑃 is contained in the set of points finitely separated from 𝑃. This is just an application of ssbl 24722 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
⊢ ∼ = (◡𝐷 “ ℝ)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ [𝑃] ∼ )
Theorem	blpnfctr 24735	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.)
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝐴 ∈ (𝑃(ball‘𝐷)+∞)) → (𝑃(ball‘𝐷)+∞) = (𝐴(ball‘𝐷)+∞))
Theorem	xmetresbl 24736	An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 24733, 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.)
⊢ 𝐵 = (𝑃(ball‘𝐷)𝑅)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))
12.4.4  Open sets of a metric space
Theorem	mopnval 24737	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 (MetOpen‘𝐷) is the family of all open sets in the metric space determined by the metric 𝐷. By mopntop 24739, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 24740. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷)))
Theorem	mopntopon 24738	The set of open sets of a metric space 𝑋 is a topology on 𝑋. 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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
Theorem	mopntop 24739	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
Theorem	mopnuni 24740	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
Theorem	elmopn 24741*	The defining property of an open set of a metric space. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ran (ball‘𝐷)(𝑥 ∈ 𝑦 ∧ 𝑦 ⊆ 𝐴))))
Theorem	mopnfss 24742	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ⊆ 𝒫 𝑋)
Theorem	mopnm 24743	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ 𝐽)
Theorem	elmopn2 24744*	A defining property of an open set of a metric space. (Contributed by NM, 5-May-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝐴 ∈ 𝐽 ↔ (𝐴 ⊆ 𝑋 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴)))
Theorem	mopnss 24745	An open set of a metric space is a subspace of its base set. (Contributed by NM, 3-Sep-2006.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋)
Theorem	isxms 24746	Express the predicate "⟨𝑋, 𝐷⟩ is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐾 ∈ TopSp ∧ 𝐽 = (MetOpen‘𝐷)))
Theorem	isxms2 24747	Express the predicate "⟨𝑋, 𝐷⟩ is an extended metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ ∞MetSp ↔ (𝐷 ∈ (∞Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
Theorem	isms 24748	Express the predicate "⟨𝑋, 𝐷⟩ is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ MetSp ↔ (𝐾 ∈ ∞MetSp ∧ 𝐷 ∈ (Met‘𝑋)))
Theorem	isms2 24749	Express the predicate "⟨𝑋, 𝐷⟩ is a metric space" with underlying set 𝑋 and distance function 𝐷. (Contributed by NM, 27-Aug-2006.) (Revised by Mario Carneiro, 24-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ MetSp ↔ (𝐷 ∈ (Met‘𝑋) ∧ 𝐽 = (MetOpen‘𝐷)))
Theorem	xmstopn 24750	The topology component of an extended metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ ∞MetSp → 𝐽 = (MetOpen‘𝐷))
Theorem	mstopn 24751	The topology component of a metric space coincides with the topology generated by the metric component. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ 𝐽 = (TopOpen‘𝐾)    &   ⊢ 𝑋 = (Base‘𝐾)    &   ⊢ 𝐷 = ((dist‘𝐾) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝐾 ∈ MetSp → 𝐽 = (MetOpen‘𝐷))
Theorem	xmstps 24752	An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
Theorem	msxms 24753	A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
Theorem	mstps 24754	A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
⊢ (𝑀 ∈ MetSp → 𝑀 ∈ TopSp)
Theorem	xmsxmet 24755	The distance function, suitably truncated, is an extended metric on 𝑋. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑀 ∈ ∞MetSp → 𝐷 ∈ (∞Met‘𝑋))
Theorem	msmet 24756	The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 12-Nov-2013.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑀 ∈ MetSp → 𝐷 ∈ (Met‘𝑋))
Theorem	msf 24757	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.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑀 ∈ MetSp → 𝐷:(𝑋 × 𝑋)⟶ℝ)
Theorem	xmsxmet2 24758	The distance function, suitably truncated, is an extended metric on 𝑋. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ (𝑀 ∈ ∞MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ (∞Met‘𝑋))
Theorem	msmet2 24759	The distance function, suitably truncated, is a metric on 𝑋. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ (𝑀 ∈ MetSp → (𝐷 ↾ (𝑋 × 𝑋)) ∈ (Met‘𝑋))
Theorem	mscl 24760	Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ)
Theorem	xmscl 24761	Closure of the distance function of an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)
Theorem	xmsge0 24762	The distance function in an extended metric space is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ≤ (𝐴𝐷𝐵))
Theorem	xmseq0 24763	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.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))
Theorem	xmssym 24764	The distance function in an extended metric space is symmetric. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
Theorem	xmstri2 24765	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
Theorem	mstri2 24766	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ (𝐶 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵)))
Theorem	xmstri 24767	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.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))
Theorem	mstri 24768	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.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐶𝐷𝐵)))
Theorem	xmstri3 24769	Triangle inequality for the distance function of an extended metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ ∞MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶)))
Theorem	mstri3 24770	Triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶)))
Theorem	msrtri 24771	Reverse triangle inequality for the distance function of a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝑋 = (Base‘𝑀)    &   ⊢ 𝐷 = (dist‘𝑀)    ⇒   ⊢ ((𝑀 ∈ MetSp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))
Theorem	xmspropd 24772	Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))    ⇒   ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))
Theorem	mspropd 24773	Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))    ⇒   ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
Theorem	setsmsbas 24774	The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 12-Nov-2024.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   ⊢ (𝜑 → 𝑋 = (Base‘𝐾))
Theorem	setsmsds 24775	The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Proof shortened by AV, 11-Nov-2024.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    ⇒   ⊢ (𝜑 → (dist‘𝑀) = (dist‘𝐾))
Theorem	setsmstset 24776	The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾))
Theorem	setsmstopn 24777	The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (MetOpen‘𝐷) = (TopOpen‘𝐾))
Theorem	setsxms 24778	The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐷 ∈ (∞Met‘𝑋)))
Theorem	setsms 24779	The constructed metric space is a metric space iff the provided distance function is a metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑋 = (Base‘𝑀))    &   ⊢ (𝜑 → 𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   ⊢ (𝜑 → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   ⊢ (𝜑 → 𝑀 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐾 ∈ MetSp ↔ 𝐷 ∈ (Met‘𝑋)))
Theorem	tmsval 24780	For any metric there is an associated metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}    &   ⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))
Theorem	tmslem 24781	Lemma for tmsbas 24782, tmsds 24783, and tmstopn 24784. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝑀 = {⟨(Base‘ndx), 𝑋⟩, ⟨(dist‘ndx), 𝐷⟩}    &   ⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑋 = (Base‘𝐾) ∧ 𝐷 = (dist‘𝐾) ∧ (MetOpen‘𝐷) = (TopOpen‘𝐾)))
Theorem	tmsbas 24782	The base set of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = (Base‘𝐾))
Theorem	tmsds 24783	The metric of a constructed metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 = (dist‘𝐾))
Theorem	tmstopn 24784	The topology of a constructed metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    &   ⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (TopOpen‘𝐾))
Theorem	tmsxms 24785	The constructed metric space is an extended metric space. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐾 ∈ ∞MetSp)
Theorem	tmsms 24786	The constructed metric space is a metric space given a metric. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐾 = (toMetSp‘𝐷)    ⇒   ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐾 ∈ MetSp)
Theorem	imasf1obl 24787	The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑍)    &   ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   ⊢ 𝐷 = (dist‘𝑈)    &   ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉))    &   ⊢ (𝜑 → 𝑃 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑆 ∈ ℝ*)    ⇒   ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆)))
Theorem	imasf1oxms 24788	The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ ∞MetSp)    ⇒   ⊢ (𝜑 → 𝑈 ∈ ∞MetSp)
Theorem	imasf1oms 24789	The image of a metric space is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅))    &   ⊢ (𝜑 → 𝑉 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ MetSp)    ⇒   ⊢ (𝜑 → 𝑈 ∈ MetSp)
Theorem	prdsbl 24790*	A ball in the product metric for finite index set is the Cartesian product of balls in all coordinates. For infinite index set this is no longer true; instead the correct statement is that a *closed ball* is the product of closed balls in each coordinate (where closed ball means a set of the form in blcld 24804) - for a counterexample the point 𝑝 in ℝ↑ℕ whose 𝑛-th coordinate is 1 − 1 / 𝑛 is in X𝑛 ∈ ℕball(0, 1) but is not in the 1-ball of the product (since 𝑑(0, 𝑝) = 1). The last assumption, 0 < 𝐴, is needed only in the case 𝐼 = ∅, when the right side evaluates to {∅} and the left evaluates to ∅ if 𝐴 ≤ 0 and {∅} if 0 < 𝐴. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅))    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑉 = (Base‘𝑅)    &   ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉))    &   ⊢ 𝐷 = (dist‘𝑌)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐼 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 0 < 𝐴)    ⇒   ⊢ (𝜑 → (𝑃(ball‘𝐷)𝐴) = X𝑥 ∈ 𝐼 ((𝑃‘𝑥)(ball‘𝐸)𝐴))
Theorem	mopni 24791*	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑥 ∧ 𝑥 ⊆ 𝐴))
Theorem	mopni2 24792*	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)
Theorem	mopni3 24793*	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝑃 ∈ 𝐴) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴))
Theorem	blssopn 24794	The balls of a metric space are open sets. (Contributed by NM, 12-Sep-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ran (ball‘𝐷) ⊆ 𝐽)
Theorem	unimopn 24795	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ⊆ 𝐽) → ∪ 𝐴 ∈ 𝐽)
Theorem	mopnin 24796	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝐽 ∧ 𝐵 ∈ 𝐽) → (𝐴 ∩ 𝐵) ∈ 𝐽)
Theorem	mopn0 24797	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.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∅ ∈ 𝐽)
Theorem	rnblopn 24798	A ball of a metric space is an open set. (Contributed by NM, 12-Sep-2006.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ ran (ball‘𝐷)) → 𝐵 ∈ 𝐽)
Theorem	blopn 24799	A ball of a metric space is an open set. (Contributed by NM, 9-Mar-2007.) (Revised by Mario Carneiro, 12-Nov-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝐽)
Theorem	neibl 24800*	The neighborhoods around a point 𝑃 of a metric space are those subsets containing a ball around 𝑃. Definition of neighborhood in [Kreyszig] p. 19. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
⊢ 𝐽 = (MetOpen‘𝐷)    ⇒   ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁 ⊆ 𝑋 ∧ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝑁)))