https://en.formulasearchengine.com/api.php?action=feedcontributions&user=131.246.17.150&feedformat=atom formulasearchengine - User contributions [en] 2021-12-08T19:44:44Z User contributions MediaWiki 1.37.0-alpha https://en.formulasearchengine.com/index.php?title=Final_topology&diff=240072 Final topology 2012-02-27T16:29:46Z <p>131.246.17.150: I doubt that a strong topology is a cat named Arthut. Undid revision 479118191 by 138.38.10.145 (talk)</p> <hr /> <div>In [[general topology]] and related areas of [[mathematics]], the '''final topology''' (or '''strong topology''' or '''colimit topology''' or '''inductive topology''') on a [[Set (mathematics)|set]] &lt;math&gt;X&lt;/math&gt;, with respect to a family of functions into &lt;math&gt;X&lt;/math&gt;, is the [[finest topology]] on ''X'' which makes those functions [[continuous function (topology)|continuous]].<br /> <br /> == Definition ==<br /> <br /> Given a set &lt;math&gt;X&lt;/math&gt; and a family of [[topological space]]s &lt;math&gt;Y_i&lt;/math&gt; with functions<br /> :&lt;math&gt;f_i: Y_i \to X&lt;/math&gt;<br /> the '''final topology''' &lt;math&gt;\tau&lt;/math&gt; on &lt;math&gt;X&lt;/math&gt; is the [[finest topology]] such that each<br /> :&lt;math&gt;f_i: Y_i \to (X,\tau)&lt;/math&gt;<br /> is [[continuous function (topology)|continuous]].<br /> <br /> Explicitly, the final topology may be described as follows: a subset ''U'' of ''X'' is open [[if and only if]] &lt;math&gt;f_i^{-1}(U)&lt;/math&gt; is open in ''Y''&lt;sub&gt;''i''&lt;/sub&gt; for each ''i'' &amp;isin; ''I''. <br /> <br /> == Examples ==<br /> <br /> * The [[quotient topology]] is the final topology on the quotient space with respect to the [[quotient map]].<br /> * The [[disjoint union (topology)|disjoint union]] is the final topology with respect to the family of [[canonical injection]]s.<br /> * More generally, a topological space is [[coherent topology|coherent]] with a family of subspaces if it has the final topology coinduced by the inclusion maps.<br /> * The [[direct limit]] of any [[direct system (mathematics)|direct system]] of spaces and continuous maps is the set-theoretic direct limit together with the final topology determined by the canonical morphisms.<br /> * Given a [[family of sets|family]] of topologies {&amp;tau;&lt;sub&gt;''i''&lt;/sub&gt;} on a fixed set ''X'' the final topology on ''X'' with respect to the functions id&lt;sub&gt;''X''&lt;/sub&gt; : (''X'', &amp;tau;&lt;sub&gt;''i''&lt;/sub&gt;) &amp;rarr; ''X'' is the [[infimum]] (or meet) of the topologies {&amp;tau;&lt;sub&gt;''i''&lt;/sub&gt;} in the [[lattice of topologies]] on ''X''. That is, the final topology &amp;tau; is the [[intersection (set theory)|intersection]] of the topologies {&amp;tau;&lt;sub&gt;''i''&lt;/sub&gt;}.<br /> <br /> == Properties ==<br /> <br /> A subset of &lt;math&gt;X&lt;/math&gt; is closed/open [[if and only if]] its preimage under ''f''&lt;sub&gt;''i''&lt;/sub&gt; is closed/open in &lt;math&gt;Y_i&lt;/math&gt; for each ''i'' &amp;isin; ''I''.<br /> <br /> The final topology on ''X'' can be characterized by the following [[universal property]]: a function &lt;math&gt;g&lt;/math&gt; from &lt;math&gt;X&lt;/math&gt; to some space &lt;math&gt;Z&lt;/math&gt; is continuous if and only if &lt;math&gt;g \circ f_i&lt;/math&gt; is continuous for each ''i'' &amp;isin; ''I''.<br /> [[Image:FinalTopology-01.png|center|Characteristic property of the final topology]]<br /> <br /> By the universal property of the [[disjoint union topology]] we know that given any family of continuous maps ''f''&lt;sub&gt;''i''&lt;/sub&gt; : ''Y''&lt;sub&gt;''i''&lt;/sub&gt; &amp;rarr; ''X'' there is a unique continuous map<br /> :&lt;math&gt;f\colon \coprod_i Y_i \to X&lt;/math&gt;<br /> If the family of maps ''f''&lt;sub&gt;''i''&lt;/sub&gt; ''covers'' ''X'' (i.e. each ''x'' in ''X'' lies in the image of some ''f''&lt;sub&gt;''i''&lt;/sub&gt;) then the map ''f'' will be a [[quotient map]] if and only if ''X'' has the final topology determined by the maps ''f''&lt;sub&gt;''i''&lt;/sub&gt;.<br /> <br /> == Categorical description ==<br /> <br /> In the language of [[category theory]], the final topology construction can be described as follows. Let ''Y'' be a [[functor]] from a [[discrete category]] ''J'' to the [[category of topological spaces]] '''Top''' which selects the spaces ''Y''&lt;sub&gt;''i''&lt;/sub&gt; for ''i'' in ''J''. Let &amp;Delta; be the [[diagonal functor]] from '''Top''' to the [[functor category]] '''Top'''&lt;sup&gt;''J''&lt;/sup&gt; (this functor sends each space ''X'' to the constant functor to ''X''). The [[comma category]] (''Y'' &amp;darr; &amp;Delta;) is then the [[category of cones]] from ''Y'', i.e. objects in (''Y'' &amp;darr; &amp;Delta;) are pairs (''X'', ''f'') where ''f''&lt;sub&gt;''i''&lt;/sub&gt; : ''Y''&lt;sub&gt;''i''&lt;/sub&gt; &amp;rarr; ''X'' is a family of continuous maps to ''X''. If ''U'' is the [[forgetful functor]] from '''Top''' to '''Set''' and &amp;Delta;&amp;prime; is the diagonal functor from '''Set''' to '''Set'''&lt;sup&gt;''J''&lt;/sup&gt; then the comma category (''UY'' &amp;darr; &amp;Delta;&amp;prime;) is the category of all cones from ''UY''. The final topology construction can then be described as a functor from (''UY'' &amp;darr; &amp;Delta;&amp;prime;) to (''Y'' &amp;darr; &amp;Delta;). This functor is [[adjoint functors|left adjoint]] to the corresponding forgetful functor.<br /> <br /> == See also ==<br /> * [[Initial topology]]<br /> <br /> ==References==<br /> * Stephen Willard, ''General Topology'', (1970) Addison-Wesley Publishing Company, Reading Massachusetts. ''(Provides a short, general introduction)''<br /> <br /> [[Category:General topology]]<br /> <br /> [[de:Finaltopologie]]<br /> [[nl:Finale topologie]]</div> 131.246.17.150