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Type | Label | Description |
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Statement | ||
Theorem | uffinfix 22101* | An ultrafilter containing a finite element is fixed. (Contributed by Jeff Hankins, 5-Dec-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
⊢ ((𝐹 ∈ (UFil‘𝑋) ∧ 𝑆 ∈ 𝐹 ∧ 𝑆 ∈ Fin) → ∃𝑥 ∈ 𝑋 𝐹 = {𝑦 ∈ 𝒫 𝑋 ∣ 𝑥 ∈ 𝑦}) | ||
Theorem | cfinufil 22102* | An ultrafilter is free iff it contains the Fréchet filter cfinfil 22067 as a subset. (Contributed by NM, 14-Jul-2008.) (Revised by Stefan O'Rear, 2-Aug-2015.) |
⊢ (𝐹 ∈ (UFil‘𝑋) → (∩ 𝐹 = ∅ ↔ {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ∈ Fin} ⊆ 𝐹)) | ||
Theorem | ufinffr 22103* | An infinite subset is contained in a free ultrafilter. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Mario Carneiro, 4-Dec-2013.) |
⊢ ((𝑋 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑋 ∧ ω ≼ 𝐴) → ∃𝑓 ∈ (UFil‘𝑋)(𝐴 ∈ 𝑓 ∧ ∩ 𝑓 = ∅)) | ||
Theorem | ufilen 22104* | Any infinite set has an ultrafilter on it whose elements are of the same cardinality as the set. Any such ultrafilter is necessarily free. (Contributed by Jeff Hankins, 7-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.) |
⊢ (ω ≼ 𝑋 → ∃𝑓 ∈ (UFil‘𝑋)∀𝑥 ∈ 𝑓 𝑥 ≈ 𝑋) | ||
Theorem | ufildr 22105 | An ultrafilter gives rise to a connected door topology. (Contributed by Jeff Hankins, 6-Dec-2009.) (Revised by Stefan O'Rear, 3-Aug-2015.) |
⊢ 𝐽 = (𝐹 ∪ {∅}) ⇒ ⊢ (𝐹 ∈ (UFil‘𝑋) → (𝐽 ∪ (Clsd‘𝐽)) = 𝒫 𝑋) | ||
Theorem | fin1aufil 22106 | There are no definable free ultrafilters in ZFC. However, there are free ultrafilters in some choice-denying constructions. Here we show that given an amorphous set (a.k.a. a Ia-finite I-infinite set) 𝑋, the set of infinite subsets of 𝑋 is a free ultrafilter on 𝑋. (Contributed by Mario Carneiro, 20-May-2015.) |
⊢ 𝐹 = (𝒫 𝑋 ∖ Fin) ⇒ ⊢ (𝑋 ∈ (FinIa ∖ Fin) → (𝐹 ∈ (UFil‘𝑋) ∧ ∩ 𝐹 = ∅)) | ||
Syntax | cfm 22107 | Extend class definition to include the neighborhood filter mapping function. |
class FilMap | ||
Syntax | cflim 22108 | Extend class notation with a function returning the limit of a filter. |
class fLim | ||
Syntax | cflf 22109 | Extend class definition to include the function for filter-based function limits. |
class fLimf | ||
Syntax | cfcls 22110 | Extend class definition to include the cluster point function on filters. |
class fClus | ||
Syntax | cfcf 22111 | Extend class definition to include the function for cluster points of a function. |
class fClusf | ||
Definition | df-fm 22112* | Define a function that takes a filter to a neighborhood filter of the range. (Since we now allow filter bases to have support smaller than the base set, the function has to come first to ensure that curryings are sets.) (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 20-Jul-2015.) |
⊢ FilMap = (𝑥 ∈ V, 𝑓 ∈ V ↦ (𝑦 ∈ (fBas‘dom 𝑓) ↦ (𝑥filGenran (𝑡 ∈ 𝑦 ↦ (𝑓 “ 𝑡))))) | ||
Definition | df-flim 22113* | Define a function (indexed by a topology 𝑗) whose value is the limits of a filter 𝑓. (Contributed by Jeff Hankins, 4-Sep-2009.) |
⊢ fLim = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ {𝑥 ∈ ∪ 𝑗 ∣ (((nei‘𝑗)‘{𝑥}) ⊆ 𝑓 ∧ 𝑓 ⊆ 𝒫 ∪ 𝑗)}) | ||
Definition | df-flf 22114* | Define a function that gives the limits of a function 𝑓 in the filter sense. (Contributed by Jeff Hankins, 14-Oct-2009.) |
⊢ fLimf = (𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑𝑚 ∪ 𝑦) ↦ (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦)))) | ||
Definition | df-fcls 22115* | Define a function that takes a filter in a topology to its set of cluster points. (Contributed by Jeff Hankins, 10-Nov-2009.) |
⊢ fClus = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ if(∪ 𝑗 = ∪ 𝑓, ∩ 𝑥 ∈ 𝑓 ((cls‘𝑗)‘𝑥), ∅)) | ||
Definition | df-fcf 22116* | Define a function that gives the cluster points of a function. (Contributed by Jeff Hankins, 24-Nov-2009.) |
⊢ fClusf = (𝑗 ∈ Top, 𝑓 ∈ ∪ ran Fil ↦ (𝑔 ∈ (∪ 𝑗 ↑𝑚 ∪ 𝑓) ↦ (𝑗 fClus ((∪ 𝑗 FilMap 𝑔)‘𝑓)))) | ||
Theorem | fmval 22117* | Introduce a function that takes a function from a filtered domain to a set and produces a filter which consists of supersets of images of filter elements. The functions which are dealt with by this function are similar to nets in topology. For example, suppose we have a sequence filtered by the filter generated by its tails under the usual positive integer ordering. Then the elements of this filter are precisely the supersets of tails of this sequence. Under this definition, it is not too difficult to see that the limit of a function in the filter sense captures the notion of convergence of a sequence. As a result, the notion of a filter generalizes many ideas associated with sequences, and this function is one way to make that relationship precise in Metamath. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = (𝑋filGenran (𝑦 ∈ 𝐵 ↦ (𝐹 “ 𝑦)))) | ||
Theorem | fmfil 22118 | A mapping filter is a filter. (Contributed by Jeff Hankins, 18-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) ∈ (Fil‘𝑋)) | ||
Theorem | fmf 22119 | Pushing-forward via a function induces a mapping on filters. (Contributed by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝐹:𝑌⟶𝑋) → (𝑋 FilMap 𝐹):(fBas‘𝑌)⟶(Fil‘𝑋)) | ||
Theorem | fmss 22120 | A finer filter produces a finer image filter. (Contributed by Jeff Hankins, 16-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐶 ∈ (fBas‘𝑌)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐵 ⊆ 𝐶)) → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ ((𝑋 FilMap 𝐹)‘𝐶)) | ||
Theorem | elfm 22121* | An element of a mapping filter. (Contributed by Jeff Hankins, 8-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 (𝐹 “ 𝑥) ⊆ 𝐴))) | ||
Theorem | elfm2 22122* | An element of a mapping filter. (Contributed by Jeff Hankins, 26-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝐿 = (𝑌filGen𝐵) ⇒ ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ (𝐴 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐿 (𝐹 “ 𝑥) ⊆ 𝐴))) | ||
Theorem | fmfg 22123 | The image filter of a filter base is the same as the image filter of its generated filter. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝐿 = (𝑌filGen𝐵) ⇒ ⊢ ((𝑋 ∈ 𝐶 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐵) = ((𝑋 FilMap 𝐹)‘𝐿)) | ||
Theorem | elfm3 22124* | An alternate formulation of elementhood in a mapping filter that requires 𝐹 to be onto. (Contributed by Jeff Hankins, 1-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝐿 = (𝑌filGen𝐵) ⇒ ⊢ ((𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌–onto→𝑋) → (𝐴 ∈ ((𝑋 FilMap 𝐹)‘𝐵) ↔ ∃𝑥 ∈ 𝐿 𝐴 = (𝐹 “ 𝑥))) | ||
Theorem | imaelfm 22125 | An image of a filter element is in the image filter. (Contributed by Jeff Hankins, 5-Oct-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝐿 = (𝑌filGen𝐵) ⇒ ⊢ (((𝑋 ∈ 𝐴 ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑆 ∈ 𝐿) → (𝐹 “ 𝑆) ∈ ((𝑋 FilMap 𝐹)‘𝐵)) | ||
Theorem | rnelfmlem 22126* | Lemma for rnelfm 22127. (Contributed by Jeff Hankins, 14-Nov-2009.) |
⊢ (((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) ∧ ran 𝐹 ∈ 𝐿) → ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥)) ∈ (fBas‘𝑌)) | ||
Theorem | rnelfm 22127 | A condition for a filter to be an image filter for a given function. (Contributed by Jeff Hankins, 14-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝑌 ∈ 𝐴 ∧ 𝐿 ∈ (Fil‘𝑋) ∧ 𝐹:𝑌⟶𝑋) → (𝐿 ∈ ran (𝑋 FilMap 𝐹) ↔ ran 𝐹 ∈ 𝐿)) | ||
Theorem | fmfnfmlem1 22128* | Lemma for fmfnfm 22132. (Contributed by Jeff Hankins, 18-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → (𝑠 ∈ (fi‘𝐵) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))) | ||
Theorem | fmfnfmlem2 22129* | Lemma for fmfnfm 22132. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → (∃𝑥 ∈ 𝐿 𝑠 = (◡𝐹 “ 𝑥) → ((𝐹 “ 𝑠) ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐿)))) | ||
Theorem | fmfnfmlem3 22130* | Lemma for fmfnfm 22132. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → (fi‘ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) = ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))) | ||
Theorem | fmfnfmlem4 22131* | Lemma for fmfnfm 22132. (Contributed by Jeff Hankins, 19-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → (𝑡 ∈ 𝐿 ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑠 ∈ (fi‘(𝐵 ∪ ran (𝑥 ∈ 𝐿 ↦ (◡𝐹 “ 𝑥))))(𝐹 “ 𝑠) ⊆ 𝑡))) | ||
Theorem | fmfnfm 22132* | A filter finer than an image filter is an image filter of the same function. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝜑 → 𝐵 ∈ (fBas‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑌⟶𝑋) & ⊢ (𝜑 → ((𝑋 FilMap 𝐹)‘𝐵) ⊆ 𝐿) ⇒ ⊢ (𝜑 → ∃𝑓 ∈ (Fil‘𝑌)(𝐵 ⊆ 𝑓 ∧ 𝐿 = ((𝑋 FilMap 𝐹)‘𝑓))) | ||
Theorem | fmufil 22133 | An image filter of an ultrafilter is an ultrafilter. (Contributed by Jeff Hankins, 11-Dec-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝑋 ∈ 𝐴 ∧ 𝐿 ∈ (UFil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (UFil‘𝑋)) | ||
Theorem | fmid 22134 | The filter map applied to the identity. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Mario Carneiro, 27-Aug-2015.) |
⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑋 FilMap ( I ↾ 𝑋))‘𝐹) = 𝐹) | ||
Theorem | fmco 22135 | Composition of image filters. (Contributed by Mario Carneiro, 27-Aug-2015.) |
⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝐵 ∈ (fBas‘𝑍)) ∧ (𝐹:𝑌⟶𝑋 ∧ 𝐺:𝑍⟶𝑌)) → ((𝑋 FilMap (𝐹 ∘ 𝐺))‘𝐵) = ((𝑋 FilMap 𝐹)‘((𝑌 FilMap 𝐺)‘𝐵))) | ||
Theorem | ufldom 22136 | The ultrafilter lemma property is a cardinal invariant, so since it transfers to subsets it also transfers over set dominance. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ ((𝑋 ∈ UFL ∧ 𝑌 ≼ 𝑋) → 𝑌 ∈ UFL) | ||
Theorem | flimval 22137* | The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil) → (𝐽 fLim 𝐹) = {𝑥 ∈ 𝑋 ∣ (((nei‘𝐽)‘{𝑥}) ⊆ 𝐹 ∧ 𝐹 ⊆ 𝒫 𝑋)}) | ||
Theorem | elflim2 22138 | The predicate "is a limit point of a filter." (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) ↔ ((𝐽 ∈ Top ∧ 𝐹 ∈ ∪ ran Fil ∧ 𝐹 ⊆ 𝒫 𝑋) ∧ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))) | ||
Theorem | flimtop 22139 | Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐽 ∈ Top) | ||
Theorem | flimneiss 22140 | A filter contains the neighborhood filter as a subfilter. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹) | ||
Theorem | flimnei 22141 | A filter contains all of the neighborhoods of its limit points. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.) |
⊢ ((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → 𝑁 ∈ 𝐹) | ||
Theorem | flimelbas 22142 | A limit point of a filter belongs to its base set. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 9-Apr-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐴 ∈ 𝑋) | ||
Theorem | flimfil 22143 | Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → 𝐹 ∈ (Fil‘𝑋)) | ||
Theorem | flimtopon 22144 | Reverse closure for the limit point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) | ||
Theorem | elflim 22145 | The predicate "is a limit point of a filter." (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹))) | ||
Theorem | flimss2 22146 | A limit point of a filter is a limit point of a finer filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐺 ⊆ 𝐹) → (𝐽 fLim 𝐺) ⊆ (𝐽 fLim 𝐹)) | ||
Theorem | flimss1 22147 | A limit point of a filter is a limit point in a coarser topology. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹)) | ||
Theorem | neiflim 22148 | A point is a limit point of its neighborhood filter. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ (𝐽 fLim ((nei‘𝐽)‘{𝐴}))) | ||
Theorem | flimopn 22149* | The condition for being a limit point of a filter still holds if one only considers open neighborhoods. (Contributed by Jeff Hankins, 4-Sep-2009.) (Proof shortened by Mario Carneiro, 9-Apr-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → 𝑥 ∈ 𝐹)))) | ||
Theorem | fbflim 22150* | A condition for a filter to converge to a point involving one of its bases. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝐹 = (𝑋filGen𝐵) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 → ∃𝑦 ∈ 𝐵 𝑦 ⊆ 𝑥)))) | ||
Theorem | fbflim2 22151* | A condition for a filter base 𝐵 to converge to a point 𝐴. Use neighborhoods instead of open neighborhoods. Compare fbflim 22150. (Contributed by FL, 4-Jul-2011.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝐹 = (𝑋filGen𝐵) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑛))) | ||
Theorem | flimclsi 22152 | The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
⊢ (𝑆 ∈ 𝐹 → (𝐽 fLim 𝐹) ⊆ ((cls‘𝐽)‘𝑆)) | ||
Theorem | hausflimlem 22153 | If 𝐴 and 𝐵 are both limits of the same filter, then all neighborhoods of 𝐴 and 𝐵 intersect. (Contributed by Mario Carneiro, 21-Sep-2015.) |
⊢ (((𝐴 ∈ (𝐽 fLim 𝐹) ∧ 𝐵 ∈ (𝐽 fLim 𝐹)) ∧ (𝑈 ∈ 𝐽 ∧ 𝑉 ∈ 𝐽) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑉)) → (𝑈 ∩ 𝑉) ≠ ∅) | ||
Theorem | hausflimi 22154* | One direction of hausflim 22155. A filter in a Hausdorff space has at most one limit. (Contributed by FL, 14-Nov-2010.) (Revised by Mario Carneiro, 21-Sep-2015.) |
⊢ (𝐽 ∈ Haus → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝐹)) | ||
Theorem | hausflim 22155* | A condition for a topology to be Hausdorff in terms of filters. A topology is Hausdorff iff every filter has at most one limit point. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))) | ||
Theorem | flimcf 22156* | Fineness is properly characterized by the property that every limit point of a filter in the finer topology is a limit point in the coarser topology. (Contributed by Jeff Hankins, 28-Sep-2009.) (Revised by Mario Carneiro, 23-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))) | ||
Theorem | flimrest 22157 | The set of limit points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐽 ↾t 𝑌) fLim (𝐹 ↾t 𝑌)) = ((𝐽 fLim 𝐹) ∩ 𝑌)) | ||
Theorem | flimclslem 22158 | Lemma for flimcls 22159. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆 ∈ 𝐹 ∧ 𝐴 ∈ (𝐽 fLim 𝐹))) | ||
Theorem | flimcls 22159* | Closure in terms of filter convergence. (Contributed by Jeff Hankins, 28-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝐴 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝑆 ∈ 𝑓 ∧ 𝐴 ∈ (𝐽 fLim 𝑓)))) | ||
Theorem | flimsncls 22160 | If 𝐴 is a limit point of the filter 𝐹, then all the points which specialize 𝐴 (in the specialization preorder) are also limit points. Thus, the set of limit points is a union of closed sets (although this is only nontrivial for non-T1 spaces). (Contributed by Mario Carneiro, 20-Sep-2015.) |
⊢ (𝐴 ∈ (𝐽 fLim 𝐹) → ((cls‘𝐽)‘{𝐴}) ⊆ (𝐽 fLim 𝐹)) | ||
Theorem | hauspwpwf1 22161* | Lemma for hauspwpwdom 22162. Points in the closure of a set in a Hausdorff space are characterized by the open neighborhoods they extend into the generating set. (Contributed by Mario Carneiro, 28-Jul-2015.) |
⊢ 𝑋 = ∪ 𝐽 & ⊢ 𝐹 = (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↦ {𝑎 ∣ ∃𝑗 ∈ 𝐽 (𝑥 ∈ 𝑗 ∧ 𝑎 = (𝑗 ∩ 𝐴))}) ⇒ ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → 𝐹:((cls‘𝐽)‘𝐴)–1-1→𝒫 𝒫 𝐴) | ||
Theorem | hauspwpwdom 22162 | If 𝑋 is a Hausdorff space, then the cardinality of the closure of a set 𝐴 is bounded by the double powerset of 𝐴. In particular, a Hausdorff space with a dense subset 𝐴 has cardinality at most 𝒫 𝒫 𝐴, and a separable Hausdorff space has cardinality at most 𝒫 𝒫 ℕ. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 28-Jul-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Haus ∧ 𝐴 ⊆ 𝑋) → ((cls‘𝐽)‘𝐴) ≼ 𝒫 𝒫 𝐴) | ||
Theorem | flffval 22163* | Given a topology and a filtered set, return the convergence function on the functions from the filtered set to the base set of the topological space. (Contributed by Jeff Hankins, 14-Oct-2009.) (Revised by Mario Carneiro, 15-Dec-2013.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑𝑚 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))) | ||
Theorem | flfval 22164 | Given a function from a filtered set to a topological space, define the set of limit points of the function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fLimf 𝐿)‘𝐹) = (𝐽 fLim ((𝑋 FilMap 𝐹)‘𝐿))) | ||
Theorem | flfnei 22165* | The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑛))) | ||
Theorem | flfneii 22166* | A neighborhood of a limit point of a function contains the image of a filter element. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (((𝐽 ∈ Top ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴})) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑁) | ||
Theorem | isflf 22167* | The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))) | ||
Theorem | flfelbas 22168 | A limit point of a function is in the topological space. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.) |
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) → 𝐴 ∈ 𝑋) | ||
Theorem | flffbas 22169* | Limit points of a function can be defined using filter bases. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.) |
⊢ 𝐿 = (𝑌filGen𝐵) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐵 (𝐹 “ 𝑠) ⊆ 𝑜)))) | ||
Theorem | flftg 22170* | Limit points of a function can be defined using topological bases. (Contributed by Mario Carneiro, 19-Sep-2015.) |
⊢ 𝐽 = (topGen‘𝐵) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐵 (𝐴 ∈ 𝑜 → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑜)))) | ||
Theorem | hausflf 22171* | If a function has its values in a Hausdorff space, then it has at most one limit value. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ∃*𝑥 𝑥 ∈ ((𝐽 fLimf 𝐿)‘𝐹)) | ||
Theorem | hausflf2 22172 | If a convergent function has its values in a Hausdorff space, then it has a unique limit. (Contributed by FL, 14-Nov-2010.) (Revised by Stefan O'Rear, 6-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (((𝐽 ∈ Haus ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ ((𝐽 fLimf 𝐿)‘𝐹) ≠ ∅) → ((𝐽 fLimf 𝐿)‘𝐹) ≈ 1o) | ||
Theorem | cnpflfi 22173 | Forward direction of cnpflf 22175. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.) |
⊢ ((𝐴 ∈ (𝐽 fLim 𝐿) ∧ 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴)) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹)) | ||
Theorem | cnpflf2 22174 | 𝐹 is continuous at point 𝐴 iff a limit of 𝐹 when 𝑥 tends to 𝐴 is (𝐹‘𝐴). Proposition 9 of [BourbakiTop1] p. TG I.50. (Contributed by FL, 29-May-2011.) (Revised by Mario Carneiro, 9-Apr-2015.) |
⊢ 𝐿 = ((nei‘𝐽)‘{𝐴}) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘𝐹)))) | ||
Theorem | cnpflf 22175* | Continuity of a function at a point in terms of filter limits. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐴 ∈ (𝐽 fLim 𝑓) → (𝐹‘𝐴) ∈ ((𝐾 fLimf 𝑓)‘𝐹))))) | ||
Theorem | cnflf 22176* | A function is continuous iff it respects filter limits. (Contributed by Jeff Hankins, 6-Sep-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fLim 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑓)‘𝐹)))) | ||
Theorem | cnflf2 22177* | A function is continuous iff it respects filter limits. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐹 “ (𝐽 fLim 𝑓)) ⊆ ((𝐾 fLimf 𝑓)‘𝐹)))) | ||
Theorem | flfcnp 22178 | A continuous function preserves filter limits. (Contributed by Mario Carneiro, 18-Sep-2015.) |
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ (𝐴 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝐺 ∈ ((𝐽 CnP 𝐾)‘𝐴))) → (𝐺‘𝐴) ∈ ((𝐾 fLimf 𝐿)‘(𝐺 ∘ 𝐹))) | ||
Theorem | lmflf 22179 | The topological limit relation on functions can be written in terms of the filter limit along the filter generated by the upper integer sets. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐿 = (𝑍filGen(ℤ≥ “ 𝑍)) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ 𝑃 ∈ ((𝐽 fLimf 𝐿)‘𝐹))) | ||
Theorem | txflf 22180* | Two sequences converge in a filter iff the sequence of their ordered pairs converges. (Contributed by Mario Carneiro, 19-Sep-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍)) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑋) & ⊢ (𝜑 → 𝐺:𝑍⟶𝑌) & ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ 〈(𝐹‘𝑛), (𝐺‘𝑛)〉) ⇒ ⊢ (𝜑 → (〈𝑅, 𝑆〉 ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘𝐻) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘𝐹) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘𝐺)))) | ||
Theorem | flfcnp2 22181* | The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 19-Sep-2015.) |
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → 𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐴))) & ⊢ (𝜑 → 𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐵))) & ⊢ (𝜑 → 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘〈𝑅, 𝑆〉)) ⇒ ⊢ (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵)))) | ||
Theorem | fclsval 22182* | The set of all cluster points of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ ((𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑌)) → (𝐽 fClus 𝐹) = if(𝑋 = 𝑌, ∩ 𝑡 ∈ 𝐹 ((cls‘𝐽)‘𝑡), ∅)) | ||
Theorem | isfcls 22183* | A cluster point of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐽 ∈ Top ∧ 𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))) | ||
Theorem | fclsfil 22184 | Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐹 ∈ (Fil‘𝑋)) | ||
Theorem | fclstop 22185 | Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐽 ∈ Top) | ||
Theorem | fclstopon 22186 | Reverse closure for the cluster point predicate. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → (𝐽 ∈ (TopOn‘𝑋) ↔ 𝐹 ∈ (Fil‘𝑋))) | ||
Theorem | isfcls2 22187* | A cluster point of a filter. (Contributed by Mario Carneiro, 26-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))) | ||
Theorem | fclsopn 22188* | Write the cluster point condition in terms of open sets. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))) | ||
Theorem | fclsopni 22189 | An open neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ (𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈 ∧ 𝑆 ∈ 𝐹)) → (𝑈 ∩ 𝑆) ≠ ∅) | ||
Theorem | fclselbas 22190 | A cluster point is in the base set. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.) |
⊢ 𝑋 = ∪ 𝐽 ⇒ ⊢ (𝐴 ∈ (𝐽 fClus 𝐹) → 𝐴 ∈ 𝑋) | ||
Theorem | fclsneii 22191 | A neighborhood of a cluster point of a filter intersects any element of that filter. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝐴 ∈ (𝐽 fClus 𝐹) ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝐴}) ∧ 𝑆 ∈ 𝐹) → (𝑁 ∩ 𝑆) ≠ ∅) | ||
Theorem | fclssscls 22192 | The set of cluster points is a subset of the closure of any filter element. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝑆 ∈ 𝐹 → (𝐽 fClus 𝐹) ⊆ ((cls‘𝐽)‘𝑆)) | ||
Theorem | fclsnei 22193* | Cluster points in terms of neighborhoods. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝐴})∀𝑠 ∈ 𝐹 (𝑛 ∩ 𝑠) ≠ ∅))) | ||
Theorem | supnfcls 22194* | The filter of supersets of 𝑋 ∖ 𝑈 does not cluster at any point of the open set 𝑈. (Contributed by Mario Carneiro, 11-Apr-2015.) (Revised by Mario Carneiro, 26-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑈 ∈ 𝐽 ∧ 𝐴 ∈ 𝑈) → ¬ 𝐴 ∈ (𝐽 fClus {𝑥 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑈) ⊆ 𝑥})) | ||
Theorem | fclsbas 22195* | Cluster points in terms of filter bases. (Contributed by Jeff Hankins, 13-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ 𝐹 = (𝑋filGen𝐵) ⇒ ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐵 (𝑜 ∩ 𝑠) ≠ ∅)))) | ||
Theorem | fclsss1 22196 | A finer topology has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝐹) ⊆ (𝐽 fClus 𝐹)) | ||
Theorem | fclsss2 22197 | A finer filter has fewer cluster points. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Mario Carneiro, 26-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐹 ⊆ 𝐺) → (𝐽 fClus 𝐺) ⊆ (𝐽 fClus 𝐹)) | ||
Theorem | fclsrest 22198 | The set of cluster points in a restricted topological space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → ((𝐽 ↾t 𝑌) fClus (𝐹 ↾t 𝑌)) = ((𝐽 fClus 𝐹) ∩ 𝑌)) | ||
Theorem | fclscf 22199* | Characterization of fineness of topologies in terms of cluster points. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))) | ||
Theorem | flimfcls 22200 | A limit point is a cluster point. (Contributed by Jeff Hankins, 12-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.) |
⊢ (𝐽 fLim 𝐹) ⊆ (𝐽 fClus 𝐹) |
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