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Another main result is that connected choice is complete for dimension greater than or equal to two in the sense that it is computably equivalent to Weak K\u0151nig\u2019s Lemma. While we can present two independent proofs for dimension three and upward that are either based on a simple geometric construction or a combinatorial argument, the proof for dimension two is based on a more involved inverse limit construction. The connected choice operation in dimension one is known to be equivalent to the Intermediate Value Theorem; we prove that this problem is not idempotent in contrast to the case of dimension two and upward. We also prove that Lipschitz continuity with Lipschitz constants strictly larger than one does not simplify finding fixed points. Finally, we prove that finding a connectedness component of a closed subset of the Euclidean unit cube of any dimension greater than or equal to one is equivalent to Weak K\u0151nig\u2019s Lemma. In order to describe these results, we introduce a representation of closed subsets of the unit cube by trees of rational complexes.<\/jats:p>","DOI":"10.1142\/s0219061319500041","type":"journal-article","created":{"date-parts":[[2018,11,28]],"date-time":"2018-11-28T07:50:24Z","timestamp":1543391424000},"page":"1950004","source":"Crossref","is-referenced-by-count":7,"title":["Connected choice and the Brouwer fixed point theorem"],"prefix":"10.1142","volume":"19","author":[{"given":"Vasco","family":"Brattka","sequence":"first","affiliation":[{"name":"Faculty of Computer Science, Universit\u00e4t der Bundeswehr M\u00fcnchen, Germany"},{"name":"Department of Mathematics and Applied Mathematics, University of Cape Town, South Africa"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"St\u00e9phane","family":"Le Roux","sequence":"additional","affiliation":[{"name":"Laboratoire Sp\u00e9cification et V\u00e9rification, CNRS &amp; \u00c9cole Normale Sup\u00e9rieure Paris-Saclay, Cachan, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Joseph S.","family":"Miller","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Wisconsin\u2013Madison, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Arno","family":"Pauly","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Swansea University, UK"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2019,6,11]]},"reference":[{"key":"S0219061319500041BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02007567"},{"key":"S0219061319500041BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-8149-3"},{"key":"S0219061319500041BIB003","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00095-4"},{"key":"S0219061319500041BIB004","doi-asserted-by":"publisher","DOI":"10.1002\/malq.200310125"},{"key":"S0219061319500041BIB005","doi-asserted-by":"publisher","DOI":"10.1137\/060658023"},{"key":"S0219061319500041BIB006","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2011.12.020"},{"key":"S0219061319500041BIB007","doi-asserted-by":"publisher","DOI":"10.1093\/logcom\/exn027"},{"key":"S0219061319500041BIB008","doi-asserted-by":"publisher","DOI":"10.2178\/bsl\/1294186663"},{"key":"S0219061319500041BIB009","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1294170993"},{"key":"S0219061319500041BIB010","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-50062-1_14"},{"key":"S0219061319500041BIB011","doi-asserted-by":"publisher","DOI":"10.1016\/j.apal.2011.10.006"},{"key":"S0219061319500041BIB013","doi-asserted-by":"publisher","DOI":"10.1007\/s00224-017-9798-1"},{"key":"S0219061319500041BIB014","doi-asserted-by":"publisher","DOI":"10.1215\/00294527-2018-0016"},{"key":"S0219061319500041BIB015","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1007\/978-3-319-40189-8_6","volume-title":"Pursuit of the Universal","volume":"9709","author":"Brattka V.","year":"2018"},{"key":"S0219061319500041BIB016","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-30870-3_7"},{"key":"S0219061319500041BIB017","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(02)00693-X"},{"key":"S0219061319500041BIB018","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3975(98)00284-9"},{"key":"S0219061319500041BIB019","doi-asserted-by":"publisher","DOI":"10.1007\/BF01456931"},{"key":"S0219061319500041BIB020","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1952.0106"},{"key":"S0219061319500041BIB021","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90001-T"},{"key":"S0219061319500041BIB022","doi-asserted-by":"publisher","DOI":"10.1016\/S0049-237X(98)80046-3"},{"key":"S0219061319500041BIB023","doi-asserted-by":"publisher","DOI":"10.1016\/j.entcs.2008.12.005"},{"key":"S0219061319500041BIB024","first-page":"107","volume-title":"CCA 2009, Proc. 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