「被覆空間」の版間の差分
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Enyokoyama (会話 | 投稿記録) →特徴: タイトルを『性質』と修正し、『共通な局所的性質』を訳出 タグ: コメントアウト |
Enyokoyama (会話 | 投稿記録) →定義: 別な定義の部分を訳出 |
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被覆空間の結果を学習する際、常に、著者の課する連結に関する前提について注意深く検討する必要がある。
また、著者によっては被覆写像が全射であることを要求しない。しかし、''X'' が連結で ''C'' が空でなければ、実際には被覆写像の全射性は他の公理から従う。
<!--== Formal definition ==
Let ''X'' be a [[topological space]]. A '''covering space''' of ''X'' is a space ''C'' together with a [[continuous function (topology)|continuous]] [[surjective]] map
:<math>p \colon C \to X\,</math>
such that for every {{nowrap|''x'' ∈ ''X''}}, there exists an [[open set|open]] [[neighborhood (topology)|neighborhood]] ''U'' of ''x'', such that ''p''<sup>−1</sup>(''U'') (the [[inverse image]] of ''U'' under ''p'') is a union of disjoint open sets in ''C'', each of which is mapped [[homeomorphism|homeomorphically]] onto ''U'' by ''p''.<ref name="Chernavskii">{{harvnb|Chernavskii|2001}}</ref><ref name="Munkres p336">{{harvnb|Munkres|2000|p=336}}</ref>
The map ''p'' is called the '''covering map''',<ref name="Munkres p336"/> the space ''X'' is often called the '''base space''' of the covering, and the space ''C'' is called the '''total space''' of the covering. For any point ''x'' in the base the inverse image of ''x'' in ''C'' is necessarily a [[discrete space]]<ref name="Munkres p336"/> called the [[Fiber (mathematics)|fiber]] over ''x''.
The special open neighborhoods ''U'' of ''x'' given in the definition are called '''evenly-covered neighborhoods'''. The evenly-covered neighborhoods form an [[open cover]] of the space ''X''. The homeomorphic copies in ''C'' of an evenly-covered neighborhood ''U'' are called the '''sheets''' over ''U''. One generally pictures ''C'' as "hovering above" ''X'', with ''p'' mapping "downwards", the sheets over ''U'' being horizontally stacked above each other and above ''U'', and the fiber over ''x'' consisting of those points of ''C'' that lie "vertically above" ''x''. In particular, covering maps are locally trivial. This means that locally, each covering map is 'isomorphic' to a projection in the sense that there is a homeomorphism, ''h'', from the pre-image ''p''<sup>−1</sup>(''U''), of an evenly covered neighbourhood ''U'', onto {{nowrap|''U'' × ''F''}}, where ''F'' is the fiber, satisfying the '''local trivialization condition''', which is that, if we project {{nowrap|''U'' × ''F''}} onto ''U'', {{nowrap|''π'' : ''U'' × ''F'' → ''U''}}, so the composition of the projection ''π'' with the homeomorphism ''h'' will be a map {{nowrap|''π'' ∘ ''h''}} from the pre-image ''p''<sup>−1</sup>(''U'') onto ''U'', then the derived composition {{nowrap|''π'' ∘ ''h''}} will equal ''p'' locally (within ''p''<sup>−1</sup>(''U'')).-->
===別な定義===
被覆写像の定義には、空間 X と C 上にある[[連結性|連結]]条件を導入することも多くある。特に、双方の空間に[[弧状連結]]と{{仮リンク|局所弧状連結|en|locally path-connected}}(locally path-connected)を要求することも多い。<ref>{{Cite book|title = An Introduction to Knot Theory|date = 1997|last = Lickorish|pages = 66–67}}</ref><ref>{{Cite book|title = Topology and Geometry|last = Bredon|year = 1997|isbn = 978-0387979267}}</ref> これを導入すると、問題の空間がこれらの性質を持つときに定理が成立することを証明することが容易にできるようになる。全射性、X が連結で C が空でないことを前提としない場合もあり、その場合は被覆写像の全射性は他の公理に従う。
<!--===Alternative definitions===
Many authors impose some [[connectedness|connectivity]] conditions on the spaces ''X'' and ''C'' in the definition of a covering map. In particular, many authors require both spaces to be [[path-connected]] and [[locally path-connected]].<ref>{{Cite book|title = An Introduction to Knot Theory|date = 1997|last = Lickorish|pages = 66–67}}</ref><ref>{{Cite book|title = Topology and Geometry|last = Bredon|year = 1997|isbn = 978-0387979267}}</ref> This can prove helpful because many theorems hold only if the spaces in question have these properties. Some authors omit the assumption of surjectivity, for if ''X'' is connected and ''C'' is nonempty then surjectivity of the covering map actually follows from the other axioms.-->
== 具体例 ==
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