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In this paper, we provide a topological semantics for belief, in particular, for Stalnaker\u2019s notion of belief defined as \u2018epistemic possibility of knowledge\u2019, in terms of the<jats:italic>closure of the interior operator<\/jats:italic>on extremally disconnected spaces. This semantics extends the standard topological interpretation of knowledge (as the interior operator) with a new topological semantics for belief. We prove that the belief logic<jats:bold>KD45<\/jats:bold>is sound and complete with respect to the class of extremally disconnected spaces and we compare our approach to a different topological setting in which belief is interpreted in terms of the derived set operator. We also study (static) belief revision as well as belief dynamics by providing a topological semantics for conditional belief and belief update modalities, respectively. Our setting based on extremally disconnected spaces, however, encounters problems when extended with dynamic updates. We then propose a solution consisting in interpreting belief in a similar way based on<jats:italic>hereditarily extremally disconnected spaces<\/jats:italic>, and axiomatize the belief logic of hereditarily extremally disconnected spaces. Finally, we provide a complete axiomatization of the logic of conditional belief and knowledge, as well as a complete axiomatization of the corresponding dynamic logic.<\/jats:p>","DOI":"10.1007\/s10992-018-9463-4","type":"journal-article","created":{"date-parts":[[2018,5,23]],"date-time":"2018-05-23T09:07:35Z","timestamp":1527066455000},"page":"205-244","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":28,"title":["A Topological Approach to Full Belief"],"prefix":"10.1007","volume":"48","author":[{"given":"Alexandru","family":"Baltag","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nick","family":"Bezhanishvili","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2523-0518","authenticated-orcid":false,"given":"Ayb\u00fcke","family":"\u00d6zg\u00fcn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sonja","family":"Smets","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2018,5,23]]},"reference":[{"issue":"1","key":"9463_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0168-0072(91)90065-T","volume":"51","author":"S Abramsky","year":"1991","unstructured":"Abramsky, S. 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