{"id":"https://openalex.org/W2010424305","doi":"https://doi.org/10.1002/malq.200710060","title":"Connectivity properties of dimension level sets","display_name":"Connectivity properties of dimension level sets","publication_year":2008,"publication_date":"2008-08-27","ids":{"openalex":"https://openalex.org/W2010424305","doi":"https://doi.org/10.1002/malq.200710060","mag":"2010424305"},"language":"en","primary_location":{"id":"doi:10.1002/malq.200710060","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.200710060","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5064796448","display_name":"Jack H. Lutz","orcid":"https://orcid.org/0000-0003-1004-3891"},"institutions":[{"id":"https://openalex.org/I173911158","display_name":"Iowa State University","ror":"https://ror.org/04rswrd78","country_code":"US","type":"education","lineage":["https://openalex.org/I173911158"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Jack H. Lutz","raw_affiliation_strings":["Department of Computer Science, Iowa State University, Ames, IA 50011, USA#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Iowa State University, Ames, IA 50011, USA#TAB#","institution_ids":["https://openalex.org/I173911158"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5050312341","display_name":"Klaus Weihrauch","orcid":null},"institutions":[{"id":"https://openalex.org/I173911158","display_name":"Iowa State University","ror":"https://ror.org/04rswrd78","country_code":"US","type":"education","lineage":["https://openalex.org/I173911158"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Klaus Weihrauch","raw_affiliation_strings":["Department of Computer Science, Iowa State University, Ames, IA 50011, USA","Department of Computer Science, Iowa State University, Ames, IA 50011, USA#TAB#"],"affiliations":[{"raw_affiliation_string":"Department of Computer Science, Iowa State University, Ames, IA 50011, USA","institution_ids":["https://openalex.org/I173911158"]},{"raw_affiliation_string":"Department of Computer Science, Iowa State University, Ames, IA 50011, USA#TAB#","institution_ids":["https://openalex.org/I173911158"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5064796448"],"corresponding_institution_ids":["https://openalex.org/I173911158"],"apc_list":{"value":2630,"currency":"USD","value_usd":2630},"apc_paid":null,"fwci":2.0026,"has_fulltext":false,"cited_by_count":19,"citation_normalized_percentile":{"value":0.86145592,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"54","issue":"5","first_page":"483","last_page":"491"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9980000257492065,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12162","display_name":"Cellular Automata and Applications","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8648838996887207},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.6252937316894531},{"id":"https://openalex.org/keywords/constructive","display_name":"Constructive","score":0.605592668056488},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5765385031700134},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.5751520395278931},{"id":"https://openalex.org/keywords/constructive-proof","display_name":"Constructive proof","score":0.52952641248703},{"id":"https://openalex.org/keywords/hausdorff-space","display_name":"Hausdorff space","score":0.523090124130249},{"id":"https://openalex.org/keywords/euclidean-distance","display_name":"Euclidean distance","score":0.43821412324905396},{"id":"https://openalex.org/keywords/totally-disconnected-space","display_name":"Totally disconnected space","score":0.43544572591781616},{"id":"https://openalex.org/keywords/hausdorff-dimension","display_name":"Hausdorff dimension","score":0.42842957377433777},{"id":"https://openalex.org/keywords/packing-dimension","display_name":"Packing dimension","score":0.4185722768306732},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3663778305053711},{"id":"https://openalex.org/keywords/minkowski\u2013bouligand-dimension","display_name":"Minkowski\u2013Bouligand dimension","score":0.3140556812286377},{"id":"https://openalex.org/keywords/locally-compact-space","display_name":"Locally compact space","score":0.16058886051177979},{"id":"https://openalex.org/keywords/fractal-dimension","display_name":"Fractal dimension","score":0.13066312670707703},{"id":"https://openalex.org/keywords/fractal","display_name":"Fractal","score":0.09775665402412415},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09475323557853699},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.08674535155296326},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.05760243535041809}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8648838996887207},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.6252937316894531},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.605592668056488},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5765385031700134},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.5751520395278931},{"id":"https://openalex.org/C202854965","wikidata":"https://www.wikidata.org/wiki/Q3044470","display_name":"Constructive proof","level":2,"score":0.52952641248703},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.523090124130249},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.43821412324905396},{"id":"https://openalex.org/C89436406","wikidata":"https://www.wikidata.org/wiki/Q1544261","display_name":"Totally disconnected space","level":3,"score":0.43544572591781616},{"id":"https://openalex.org/C194198291","wikidata":"https://www.wikidata.org/wiki/Q565186","display_name":"Hausdorff dimension","level":2,"score":0.42842957377433777},{"id":"https://openalex.org/C80661125","wikidata":"https://www.wikidata.org/wiki/Q5404299","display_name":"Packing dimension","level":5,"score":0.4185722768306732},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3663778305053711},{"id":"https://openalex.org/C101597101","wikidata":"https://www.wikidata.org/wiki/Q2555655","display_name":"Minkowski\u2013Bouligand dimension","level":4,"score":0.3140556812286377},{"id":"https://openalex.org/C31498916","wikidata":"https://www.wikidata.org/wiki/Q583034","display_name":"Locally compact space","level":2,"score":0.16058886051177979},{"id":"https://openalex.org/C26546657","wikidata":"https://www.wikidata.org/wiki/Q1412452","display_name":"Fractal dimension","level":3,"score":0.13066312670707703},{"id":"https://openalex.org/C40636538","wikidata":"https://www.wikidata.org/wiki/Q81392","display_name":"Fractal","level":2,"score":0.09775665402412415},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09475323557853699},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.08674535155296326},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.05760243535041809},{"id":"https://openalex.org/C98045186","wikidata":"https://www.wikidata.org/wiki/Q205663","display_name":"Process (computing)","level":2,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/malq.200710060","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.200710060","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W109020964","https://openalex.org/W1490869128","https://openalex.org/W1594643737","https://openalex.org/W1595391242","https://openalex.org/W1664039392","https://openalex.org/W1966996896","https://openalex.org/W1978306213","https://openalex.org/W1979604519","https://openalex.org/W1985783244","https://openalex.org/W2032121576","https://openalex.org/W2075508378","https://openalex.org/W2076165639","https://openalex.org/W2118349518","https://openalex.org/W2148932839","https://openalex.org/W2160064439","https://openalex.org/W2939440802","https://openalex.org/W4230960895","https://openalex.org/W4238652859","https://openalex.org/W4242145395","https://openalex.org/W4245497173"],"related_works":["https://openalex.org/W2054104090","https://openalex.org/W2019545391","https://openalex.org/W2134178586","https://openalex.org/W2962871033","https://openalex.org/W2912352481","https://openalex.org/W1969747272","https://openalex.org/W2259419819","https://openalex.org/W1991044401","https://openalex.org/W2389584428","https://openalex.org/W2564599082"],"abstract_inverted_index":{"Abstract":[0],"This":[1],"paper":[2],"initiates":[3],"the":[4,23,41,45,104,133],"study":[5],"of":[6,22,25,44,51,66,74,145,151],"sets":[7,105,134],"in":[8,20,40,54,62,83,154],"Euclidean":[9,155],"spaces":[10],"\u211d":[11,55],"n":[12,14,35,56,93,96,127,130],"(":[13,92],"\u2265":[15],"2)":[16],"that":[17,103,119],"are":[18,38,112,141],"defined":[19],"terms":[21],"dimensions":[24,60,81],"their":[26],"elements.":[27],"Specifically,":[28],"given":[29],"an":[30],"interval":[31],"I":[32,48,63,70,84,87,107,110,121,136,139],"\u2286":[33],"[0,":[34,89,123],"],":[36,97,131],"we":[37,117],"interested":[39],"connectivity":[42],"properties":[43,150],"set":[46],"DIM":[47,69,106,109,135,138],",":[49,64,72],"consisting":[50,73],"all":[52,75],"points":[53,76],"whose":[57,77],"(constructive":[58,79],"Hausdorff)":[59],"lie":[61,82],"and":[65,108,137],"its":[67],"dual":[68],"str":[71,111,140],"strong":[78],"packing)":[80],".":[85],"If":[86],"is":[88,99,122],"1)":[90],"or":[91,125],"\u2013":[94,128],"1,":[95,129],"it":[98],"easy":[100],"to":[101],"see":[102],"totally":[113],"disconnected.":[114],"In":[115],"contrast,":[116],"show":[118],"if":[120],"1]":[124],"[":[126],"then":[132],"path\u2010connected.":[142],"Our":[143],"proof":[144],"this":[146],"fact":[147],"uses":[148],"geometric":[149],"Kolmogorov":[152],"complexity":[153],"spaces.":[156],"(\u00a9":[157],"2008":[158],"WILEY\u2010VCH":[159],"Verlag":[160],"GmbH":[161],"&amp;":[162],"Co.":[163],"KGaA,":[164],"Weinheim)":[165]},"counts_by_year":[{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":2}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}
