{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T01:06:58Z","timestamp":1783904818804,"version":"3.55.0"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2014,3,30]],"date-time":"2014-03-30T00:00:00Z","timestamp":1396137600000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-sa\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    In this article we focus on a special case of the Brouwer invariance of domain theorem. Let us A, B be a subsets of \u03b5\n                    <jats:sup>n<\/jats:sup>\n                    , and f : A \u2192 B be a homeomorphic. We prove that, if A is closed then f transform the boundary of A to the boundary of B; and if B is closed then f transform the interior of A to the interior of B. These two cases are sufficient to prove the topological invariance of dimension, which is used to prove basic properties of the n-dimensional manifolds, and also to prove basic properties of the boundary and the interior of manifolds, e.g. the boundary of an n-dimension manifold with boundary is an (n \u2212 1)-dimension manifold. This article is based on [18]; [21] and [20] can also serve as reference books.\n                  <\/jats:p>","DOI":"10.2478\/forma-2014-0003","type":"journal-article","created":{"date-parts":[[2015,5,28]],"date-time":"2015-05-28T16:26:02Z","timestamp":1432830362000},"page":"21-28","source":"Crossref","is-referenced-by-count":1,"title":["Brouwer Invariance of Domain Theorem"],"prefix":"10.2478","volume":"22","author":[{"given":"Karol","family":"P\u0105k","sequence":"first","affiliation":[{"name":"Institute of Informatics University of Bia\u0142ystok Sosnowa 64, 15-887 Bia\u0142ystok Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2014,3,30]]},"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/22\/1\/article-p21.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2014-0003","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T00:17:50Z","timestamp":1783901870000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2014-0003"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,3,30]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2014,3,30]]}},"alternative-id":["10.2478\/forma-2014-0003"],"URL":"https:\/\/doi.org\/10.2478\/forma-2014-0003","relation":{},"ISSN":["1898-9934"],"issn-type":[{"value":"1898-9934","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,3,30]]}}}