HomeHome Intuitionistic Logic Explorer
Theorem List (p. 131 of 139)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  ILE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Theorem List for Intuitionistic Logic Explorer - 13001-13100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremblres 13001 A ball in a restricted metric space. (Contributed by Mario Carneiro, 5-Jan-2014.)
𝐶 = (𝐷 ↾ (𝑌 × 𝑌))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ (𝑋𝑌) ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝐶)𝑅) = ((𝑃(ball‘𝐷)𝑅) ∩ 𝑌))
 
Theoremxmeterval 13002 Value of the "finitely separated" relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       (𝐷 ∈ (∞Met‘𝑋) → (𝐴 𝐵 ↔ (𝐴𝑋𝐵𝑋 ∧ (𝐴𝐷𝐵) ∈ ℝ)))
 
Theoremxmeter 13003 The "finitely separated" relation is an equivalence relation. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       (𝐷 ∈ (∞Met‘𝑋) → Er 𝑋)
 
Theoremxmetec 13004 The equivalence classes under the finite separation equivalence relation are infinity balls. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) → [𝑃] = (𝑃(ball‘𝐷)+∞))
 
Theoremblssec 13005 A ball centered at 𝑃 is contained in the set of points finitely separated from 𝑃. This is just an application of ssbl 12993 to the infinity ball. (Contributed by Mario Carneiro, 24-Aug-2015.)
= (𝐷 “ ℝ)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑆 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑆) ⊆ [𝑃] )
 
Theoremblpnfctr 13006 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) ∧ 𝑃𝑋𝐴 ∈ (𝑃(ball‘𝐷)+∞)) → (𝑃(ball‘𝐷)+∞) = (𝐴(ball‘𝐷)+∞))
 
Theoremxmetresbl 13007 An extended metric restricted to any ball (in particular the infinity ball) is a proper metric. Together with xmetec 13004, 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝐷 ↾ (𝐵 × 𝐵)) ∈ (Met‘𝐵))
 
7.2.4  Open sets of a metric space
 
Theoremmopnrel 13008 The class of open sets of a metric space is a relation. (Contributed by Jim Kingdon, 5-May-2023.)
Rel MetOpen
 
Theoremmopnval 13009 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 13011, the open sets of a metric space form a topology 𝐽, whose base set is 𝐽 by mopnuni 13012. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷)))
 
Theoremmopntopon 13010 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))
 
Theoremmopntop 13011 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)
 
Theoremmopnuni 13012 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)
 
Theoremelmopn 13013* 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) → (𝐴𝐽 ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑦 ∈ ran (ball‘𝐷)(𝑥𝑦𝑦𝐴))))
 
Theoremmopnfss 13014 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)
 
Theoremmopnm 13015 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) → 𝑋𝐽)
 
Theoremelmopn2 13016* 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) → (𝐴𝐽 ↔ (𝐴𝑋 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+ (𝑥(ball‘𝐷)𝑦) ⊆ 𝐴)))
 
Theoremmopnss 13017 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) ∧ 𝐴𝐽) → 𝐴𝑋)
 
Theoremisxms 13018 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)))
 
Theoremisxms2 13019 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)))
 
Theoremisms 13020 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)))
 
Theoremisms2 13021 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)))
 
Theoremxmstopn 13022 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))
 
Theoremmstopn 13023 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))
 
Theoremxmstps 13024 An extended metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ ∞MetSp → 𝑀 ∈ TopSp)
 
Theoremmsxms 13025 A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp)
 
Theoremmstps 13026 A metric space is a topological space. (Contributed by Mario Carneiro, 26-Aug-2015.)
(𝑀 ∈ MetSp → 𝑀 ∈ TopSp)
 
Theoremxmsxmet 13027 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))
 
Theoremmsmet 13028 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))
 
Theoremmsf 13029 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)
 
Theoremxmsxmet2 13030 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))
 
Theoremmsmet2 13031 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))
 
Theoremmscl 13032 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ)
 
Theoremxmscl 13033 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) ∈ ℝ*)
 
Theoremxmsge0 13034 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 ∧ 𝐴𝑋𝐵𝑋) → 0 ≤ (𝐴𝐷𝐵))
 
Theoremxmseq0 13035 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 ∧ 𝐴𝑋𝐵𝑋) → ((𝐴𝐷𝐵) = 0 ↔ 𝐴 = 𝐵))
 
Theoremxmssym 13036 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 ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝐵𝐷𝐴))
 
Theoremxmstri2 13037 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 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) +𝑒 (𝐶𝐷𝐵)))
 
Theoremmstri2 13038 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 ∧ (𝐶𝑋𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐶𝐷𝐴) + (𝐶𝐷𝐵)))
 
Theoremxmstri 13039 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐶𝐷𝐵)))
 
Theoremmstri 13040 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐶𝐷𝐵)))
 
Theoremxmstri3 13041 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) +𝑒 (𝐵𝐷𝐶)))
 
Theoremmstri3 13042 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐷𝐵) ≤ ((𝐴𝐷𝐶) + (𝐵𝐷𝐶)))
 
Theoremmsrtri 13043 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 ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (abs‘((𝐴𝐷𝐶) − (𝐵𝐷𝐶))) ≤ (𝐴𝐷𝐵))
 
Theoremxmspropd 13044 Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ ∞MetSp ↔ 𝐿 ∈ ∞MetSp))
 
Theoremmspropd 13045 Property deduction for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ MetSp ↔ 𝐿 ∈ MetSp))
 
Theoremsetsmsbasg 13046 The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)    &   (𝜑 → (MetOpen‘𝐷) ∈ 𝑊)       (𝜑𝑋 = (Base‘𝐾))
 
Theoremsetsmsdsg 13047 The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)    &   (𝜑 → (MetOpen‘𝐷) ∈ 𝑊)       (𝜑 → (dist‘𝑀) = (dist‘𝐾))
 
Theoremsetsmstsetg 13048 The topology of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) (Revised by Jim Kingdon, 7-May-2023.)
(𝜑𝑋 = (Base‘𝑀))    &   (𝜑𝐷 = ((dist‘𝑀) ↾ (𝑋 × 𝑋)))    &   (𝜑𝐾 = (𝑀 sSet ⟨(TopSet‘ndx), (MetOpen‘𝐷)⟩))    &   (𝜑𝑀𝑉)    &   (𝜑 → (MetOpen‘𝐷) ∈ 𝑊)       (𝜑 → (MetOpen‘𝐷) = (TopSet‘𝐾))
 
Theoremmopni 13049* 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) ∧ 𝐴𝐽𝑃𝐴) → ∃𝑥 ∈ ran (ball‘𝐷)(𝑃𝑥𝑥𝐴))
 
Theoremmopni2 13050* 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) ∧ 𝐴𝐽𝑃𝐴) → ∃𝑥 ∈ ℝ+ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴)
 
Theoremmopni3 13051* 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) ∧ 𝐴𝐽𝑃𝐴) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ ℝ+ (𝑥 < 𝑅 ∧ (𝑃(ball‘𝐷)𝑥) ⊆ 𝐴))
 
Theoremblssopn 13052 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)
 
Theoremunimopn 13053 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) ∧ 𝐴𝐽) → 𝐴𝐽)
 
Theoremmopnin 13054 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) ∧ 𝐴𝐽𝐵𝐽) → (𝐴𝐵) ∈ 𝐽)
 
Theoremmopn0 13055 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)
 
Theoremrnblopn 13056 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)) → 𝐵𝐽)
 
Theoremblopn 13057 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) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝐷)𝑅) ∈ 𝐽)
 
Theoremneibl 13058* 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) ∧ 𝑃𝑋) → (𝑁 ∈ ((nei‘𝐽)‘{𝑃}) ↔ (𝑁𝑋 ∧ ∃𝑟 ∈ ℝ+ (𝑃(ball‘𝐷)𝑟) ⊆ 𝑁)))
 
Theoremblnei 13059 A ball around a point is a neighborhood of the point. (Contributed by NM, 8-Nov-2007.) (Revised by Mario Carneiro, 24-Aug-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) ∈ ((nei‘𝐽)‘{𝑃}))
 
Theoremblsscls2 13060* A smaller closed ball is contained in a larger open ball. (Contributed by Mario Carneiro, 10-Jan-2014.)
𝐽 = (MetOpen‘𝐷)    &   𝑆 = {𝑧𝑋 ∣ (𝑃𝐷𝑧) ≤ 𝑅}       (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋) ∧ (𝑅 ∈ ℝ*𝑇 ∈ ℝ*𝑅 < 𝑇)) → 𝑆 ⊆ (𝑃(ball‘𝐷)𝑇))
 
Theoremmetss 13061* Two ways of saying that metric 𝐷 generates a finer topology than metric 𝐶. (Contributed by Mario Carneiro, 12-Nov-2013.) (Revised by Mario Carneiro, 24-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽𝐾 ↔ ∀𝑥𝑋𝑟 ∈ ℝ+𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟)))
 
Theoremmetequiv 13062* 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.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (𝐽 = 𝐾 ↔ ∀𝑥𝑋 (∀𝑟 ∈ ℝ+𝑠 ∈ ℝ+ (𝑥(ball‘𝐷)𝑠) ⊆ (𝑥(ball‘𝐶)𝑟) ∧ ∀𝑎 ∈ ℝ+𝑏 ∈ ℝ+ (𝑥(ball‘𝐶)𝑏) ⊆ (𝑥(ball‘𝐷)𝑎))))
 
Theoremmetequiv2 13063* 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.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑋)) → (∀𝑥𝑋𝑟 ∈ ℝ+𝑠 ∈ ℝ+ (𝑠𝑟 ∧ (𝑥(ball‘𝐶)𝑠) = (𝑥(ball‘𝐷)𝑠)) → 𝐽 = 𝐾))
 
Theoremmetss2lem 13064* Lemma for metss2 13065. (Contributed by Mario Carneiro, 14-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   (𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       ((𝜑 ∧ (𝑥𝑋𝑆 ∈ ℝ+)) → (𝑥(ball‘𝐷)(𝑆 / 𝑅)) ⊆ (𝑥(ball‘𝐶)𝑆))
 
Theoremmetss2 13065* If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), then 𝐷 generates a finer topology. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they generate the same topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   (𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       (𝜑𝐽𝐾)
 
Theoremcomet 13066* The composition of an extended metric with a monotonic subadditive function is an extended metric. (Contributed by Mario Carneiro, 21-Mar-2015.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐹:(0[,]+∞)⟶ℝ*)    &   ((𝜑𝑥 ∈ (0[,]+∞)) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0))    &   ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝑥𝑦 → (𝐹𝑥) ≤ (𝐹𝑦)))    &   ((𝜑 ∧ (𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞))) → (𝐹‘(𝑥 +𝑒 𝑦)) ≤ ((𝐹𝑥) +𝑒 (𝐹𝑦)))       (𝜑 → (𝐹𝐷) ∈ (∞Met‘𝑋))
 
Theorembdmetval 13067* Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))       (((𝐶:(𝑋 × 𝑋)⟶ℝ*𝑅 ∈ ℝ*) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ*, < ))
 
Theorembdxmet 13068* The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐷 ∈ (∞Met‘𝑋))
 
Theorembdmet 13069* The standard bounded metric is a proper metric given an extended metric and a positive real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ+) → 𝐷 ∈ (Met‘𝑋))
 
Theorembdbl 13070* The standard bounded metric corresponding to 𝐶 generates the same balls as 𝐶 for radii less than 𝑅. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) ∧ (𝑃𝑋𝑆 ∈ ℝ*𝑆𝑅)) → (𝑃(ball‘𝐷)𝑆) = (𝑃(ball‘𝐶)𝑆))
 
Theorembdmopn 13071* The standard bounded metric corresponding to 𝐶 generates the same topology as 𝐶. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 19-May-2023.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ*, < ))    &   𝐽 = (MetOpen‘𝐶)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ* ∧ 0 < 𝑅) → 𝐽 = (MetOpen‘𝐷))
 
Theoremmopnex 13072* The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (∞Met‘𝑋) → ∃𝑑 ∈ (Met‘𝑋)𝐽 = (MetOpen‘𝑑))
 
Theoremmetrest 13073 Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)
𝐷 = (𝐶 ↾ (𝑌 × 𝑌))    &   𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) = 𝐾)
 
Theoremxmetxp 13074* The maximum metric (Chebyshev distance) on the product of two sets. (Contributed by Jim Kingdon, 11-Oct-2023.)
𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))       (𝜑𝑃 ∈ (∞Met‘(𝑋 × 𝑌)))
 
Theoremxmetxpbl 13075* The maximum metric (Chebyshev distance) on the product of two sets, expressed in terms of balls centered on a point 𝐶 with radius 𝑅. (Contributed by Jim Kingdon, 22-Oct-2023.)
𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   (𝜑𝑅 ∈ ℝ*)    &   (𝜑𝐶 ∈ (𝑋 × 𝑌))       (𝜑 → (𝐶(ball‘𝑃)𝑅) = (((1st𝐶)(ball‘𝑀)𝑅) × ((2nd𝐶)(ball‘𝑁)𝑅)))
 
Theoremxmettxlem 13076* Lemma for xmettx 13077. (Contributed by Jim Kingdon, 15-Oct-2023.)
𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   𝐽 = (MetOpen‘𝑀)    &   𝐾 = (MetOpen‘𝑁)    &   𝐿 = (MetOpen‘𝑃)       (𝜑𝐿 ⊆ (𝐽 ×t 𝐾))
 
Theoremxmettx 13077* The maximum metric (Chebyshev distance) on the product of two sets, expressed as a binary topological product. (Contributed by Jim Kingdon, 11-Oct-2023.)
𝑃 = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ sup({((1st𝑢)𝑀(1st𝑣)), ((2nd𝑢)𝑁(2nd𝑣))}, ℝ*, < ))    &   (𝜑𝑀 ∈ (∞Met‘𝑋))    &   (𝜑𝑁 ∈ (∞Met‘𝑌))    &   𝐽 = (MetOpen‘𝑀)    &   𝐾 = (MetOpen‘𝑁)    &   𝐿 = (MetOpen‘𝑃)       (𝜑𝐿 = (𝐽 ×t 𝐾))
 
7.2.5  Continuity in metric spaces
 
Theoremmetcnp3 13078* Two ways to express that 𝐹 is continuous at 𝑃 for metric spaces. Proposition 14-4.2 of [Gleason] p. 240. (Contributed by NM, 17-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+ (𝐹 “ (𝑃(ball‘𝐶)𝑧)) ⊆ ((𝐹𝑃)(ball‘𝐷)𝑦))))
 
Theoremmetcnp 13079* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. (Contributed by NM, 11-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹𝑃)𝐷(𝐹𝑤)) < 𝑦))))
 
Theoremmetcnp2 13080* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. The distance arguments are swapped compared to metcnp 13079 (and Munkres' metcn 13081) for compatibility with df-lm 12757. Definition 1.3-3 of [Kreyszig] p. 20. (Contributed by NM, 4-Jun-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝑃𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑤𝐶𝑃) < 𝑧 → ((𝐹𝑤)𝐷(𝐹𝑃)) < 𝑦))))
 
Theoremmetcn 13081* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous. Theorem 10.1 of [Munkres] p. 127. The second biconditional argument says that for every positive "epsilon" 𝑦 there is a positive "delta" 𝑧 such that a distance less than delta in 𝐶 maps to a distance less than epsilon in 𝐷. (Contributed by NM, 15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝐹𝑥)𝐷(𝐹𝑤)) < 𝑦))))
 
Theoremmetcnpi 13082* Epsilon-delta property of a continuous metric space function, with function arguments as in metcnp 13079. (Contributed by NM, 17-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+𝑦𝑋 ((𝑃𝐶𝑦) < 𝑥 → ((𝐹𝑃)𝐷(𝐹𝑦)) < 𝐴))
 
Theoremmetcnpi2 13083* Epsilon-delta property of a continuous metric space function, with swapped distance function arguments as in metcnp2 13080. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+𝑦𝑋 ((𝑦𝐶𝑃) < 𝑥 → ((𝐹𝑦)𝐷(𝐹𝑃)) < 𝐴))
 
Theoremmetcnpi3 13084* Epsilon-delta property of a metric space function continuous at 𝑃. A variation of metcnpi2 13083 with non-strict ordering. (Contributed by NM, 16-Dec-2007.) (Revised by Mario Carneiro, 13-Nov-2013.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) ∧ (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ∧ 𝐴 ∈ ℝ+)) → ∃𝑥 ∈ ℝ+𝑦𝑋 ((𝑦𝐶𝑃) ≤ 𝑥 → ((𝐹𝑦)𝐷(𝐹𝑃)) ≤ 𝐴))
 
Theoremtxmetcnp 13085* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by Jim Kingdon, 22-Oct-2023.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (MetOpen‘𝐸)       (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) ∧ (𝐴𝑋𝐵𝑌)) → (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((𝐴𝐶𝑢) < 𝑤 ∧ (𝐵𝐷𝑣) < 𝑤) → ((𝐴𝐹𝐵)𝐸(𝑢𝐹𝑣)) < 𝑧))))
 
Theoremtxmetcn 13086* Continuity of a binary operation on metric spaces. (Contributed by Mario Carneiro, 2-Sep-2015.)
𝐽 = (MetOpen‘𝐶)    &   𝐾 = (MetOpen‘𝐷)    &   𝐿 = (MetOpen‘𝐸)       ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌) ∧ 𝐸 ∈ (∞Met‘𝑍)) → (𝐹 ∈ ((𝐽 ×t 𝐾) Cn 𝐿) ↔ (𝐹:(𝑋 × 𝑌)⟶𝑍 ∧ ∀𝑥𝑋𝑦𝑌𝑧 ∈ ℝ+𝑤 ∈ ℝ+𝑢𝑋𝑣𝑌 (((𝑥𝐶𝑢) < 𝑤 ∧ (𝑦𝐷𝑣) < 𝑤) → ((𝑥𝐹𝑦)𝐸(𝑢𝐹𝑣)) < 𝑧))))
 
Theoremmetcnpd 13087* Two ways to say a mapping from metric 𝐶 to metric 𝐷 is continuous at point 𝑃. (Contributed by Jim Kingdon, 14-Jun-2023.)
(𝜑𝐽 = (MetOpen‘𝐶))    &   (𝜑𝐾 = (MetOpen‘𝐷))    &   (𝜑𝐶 ∈ (∞Met‘𝑋))    &   (𝜑𝐷 ∈ (∞Met‘𝑌))    &   (𝜑𝑃𝑋)       (𝜑 → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑃) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑃𝐶𝑤) < 𝑧 → ((𝐹𝑃)𝐷(𝐹𝑤)) < 𝑦))))
 
7.2.6  Topology on the reals
 
Theoremqtopbasss 13088* The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.) (Revised by Jim Kingdon, 22-May-2023.)
𝑆 ⊆ ℝ*    &   ((𝑥𝑆𝑦𝑆) → sup({𝑥, 𝑦}, ℝ*, < ) ∈ 𝑆)    &   ((𝑥𝑆𝑦𝑆) → inf({𝑥, 𝑦}, ℝ*, < ) ∈ 𝑆)       ((,) “ (𝑆 × 𝑆)) ∈ TopBases
 
Theoremqtopbas 13089 The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
((,) “ (ℚ × ℚ)) ∈ TopBases
 
Theoremretopbas 13090 A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
ran (,) ∈ TopBases
 
Theoremretop 13091 The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
(topGen‘ran (,)) ∈ Top
 
Theoremuniretop 13092 The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
ℝ = (topGen‘ran (,))
 
Theoremretopon 13093 The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
(topGen‘ran (,)) ∈ (TopOn‘ℝ)
 
Theoremretps 13094 The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
𝐾 = {⟨(Base‘ndx), ℝ⟩, ⟨(TopSet‘ndx), (topGen‘ran (,))⟩}       𝐾 ∈ TopSp
 
Theoremiooretopg 13095 Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.) (Revised by Jim Kingdon, 23-May-2023.)
((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
 
Theoremcnmetdval 13096 Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
𝐷 = (abs ∘ − )       ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴𝐵)))
 
Theoremcnmet 13097 The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
(abs ∘ − ) ∈ (Met‘ℂ)
 
Theoremcnxmet 13098 The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
(abs ∘ − ) ∈ (∞Met‘ℂ)
 
Theoremcntoptopon 13099 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))       𝐽 ∈ (TopOn‘ℂ)
 
Theoremcntoptop 13100 The topology of the complex numbers is a topology. (Contributed by Jim Kingdon, 6-Jun-2023.)
𝐽 = (MetOpen‘(abs ∘ − ))       𝐽 ∈ Top
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13821
  Copyright terms: Public domain < Previous  Next >