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{{short description|Equivalence of notions of density for sets of multiples of integers}} In [[number theory]], the '''Davenport–Erdős theorem''' states that, for sets of multiples of integers, several different notions of [[Natural density|de ...

...\}</math> is not [[closed (topology)|closed]] since it doesn't contain its limit point 0. Relative to the K-topology however, the set ''K'' is declared to b ...the collection of all open intervals <math>(a,b)</math> together with all sets of the form <math>(a,b)\setminus K.</math>{{sfn|Munkres|2000|p=82}} ...

{{short description|On decreasing nested sequences of non-empty compact sets}} ...rsections of decreasing nested [[sequence|sequences]] of non-empty compact sets. ...

...r bound 0, then the sequence <math>m(x_n)</math> has [[Limit (mathematics)|limit]]&nbsp;0. ...ding Boolean algebra of [[measurable set]]s modulo [[Null set|measure zero sets]] is complete, it is a Maharam algebra. ...

...number]]s {''x_k''} with the property that {''x_k''} [[Limit of a sequence|converge]]s to ''x'' with respect to the [[Euclidean topology ...and using as a [[neighborhood base]] for each irrational number ''x'', the sets <math> U_n(x) = \{ x_k : k \ge n \} \cup \{x\}.</math> ...

...math> is given the [[Neighbourhood system|local basis]] of relatively open sets inherited from the [[Euclidean topology]] on <math>(0,1)^2</math>. The rema ..., respectively. Since <math>r\sqrt{2}<\min\{1/n,1/m\}</math>, these closed sets containing <math>U_n(0,0)</math> and <math>U_k(1/2,r\sqrt{2})</math> for so ...

...cover <math>\mathcal U</math>, to allow the pairwise intersections of the sets in <math>\mathcal U=\mathcal U_0</math> to be covered by an open cover <mat ...\left(\operatorname{\mathbf{cosk}}_n U_\bullet\right)_{n+1}</math> is the limit of the diagram which has one copy of <math>U_i</math> for each <math>i</mat ...

=== Definition for sets === ..._2</math> is approximately equal to two straight lines that overlap in the limit. It would be reasonable to say it has an approximate tangent space <math>\m ...

{{short description|Mathematical problem concerning limit cycles in dynamical systems}} {{unsolved|mathematics|Is there a [[uniformly bounded|uniform bound]] on [[limit cycles]] in generic finite-parameter families of [[vector fields]] on a sph ...

* [[Limit set]] [[Category:Limit sets]] ...

...e these are dense {{math|''G''_''δ''}} sets, the zero set of the limit function is also dense. ...'' a Pompeiu derivative assumes both positive and negative values in dense sets, in the precise meaning that such functions constitute a residual set of th ...

...ility notation deals with [[Convergence of random variables|convergence of sets of random variables]], where convergence is in the sense of [[convergence i ...sub>'' converges to zero in probability as ''n'' approaches an appropriate limit. ...

...p_{j \in \N} D_{ij}</math> [[Absorbing set|absorbs]] <math>D_i.</math> The sets <math>\left(D_{ij}\right)_{i,j \in \N}</math> will form the second stratum. ...\in \N} D_{ijk}</math> [[Absorbing set|absorbs]] <math>D_{ij}.</math> The sets <math>\left(D_{ijk}\right)_{i,j,k \in \N}</math> form the third stratum. ...

...ome [[cardinal number]] <math>\kappa</math> can be written as the union of sets <math>X_1, X_2,...</math> where <math>X_n</math> is of [[order type]] at mo The proof is by transfinite induction. Let ''<math>\alpha</math>'' be a limit ordinal (the induction is trivial for successor ordinals), and for each ''< ...

...th>\tau</math> is extended to a topology <math>\sigma</math> by adding the sets of the form ...gle in the topology <math>\tau</math> is a regular open set and so are the sets <math>U(\alpha,n)</math> defined above with which the topology was expanded ...

...line before article has been created. -->{{Not to be confused with|Direct limit of groups}} ...ref><ref>{{Cite journal |last=Groves |first=Daniel |date=2009-07-26 |title=Limit groups for relatively hyperbolic groups. I. The basic tools |url=https://ms ...

...nications]] the '''Chu–Harrington limit''' or '''Chu limit''' sets a lower limit on the [[Q factor]] for a small [[radio antenna]].<ref name="Bing">{{cite b | url = https://books.google.com/books?id=L__jFgCNJuEC&dq=chu-harrington+limit&pg=PA567 ...

In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almo * Every point of ''X'' has a [[local basis]] of [[clopen sets]], i.e., ''X'' is a [[zero-dimensional space]].<ref name="CEIT"/><br>''Proo ...

...t membership]] serving as the order relation. <math>\omega_1</math> is a [[limit ordinal]], i.e. there is no ordinal <math>\alpha</math> such that <math>\om ...s, <math>\omega_1</math> and <math>\aleph_1</math> are considered equal as sets. To generalize: if <math>\alpha</math> is an arbitrary ordinal, we define < ...

...el axioms of set theory]] are isomorphic to a member of a certain class of sets. ...on <math>\vDash</math>, he defines a "normal domain" to be a collection of sets along with the true <math>\in</math> relation that satisfies <math>\mathrm{ ...