en>Dima373: Optics-stub

2013-06-24T08:00:58Z

Optics-stub

← Older revision		Revision as of 10:00, 24 June 2013
Line 1:		Line 1:
	~~We had 2 or 3 days off but we’re again now~~, ~~for you~~, with ~~the newest Subway People Hack! It was created for all players all over the world~~. ~~Subway Surfers is~~ a ~~game which~~ is ~~called “endless running” game. In this game you~~'~~re young hooligan who painted graffiti on~~ the ~~wall~~. ~~For this reason you should escape from~~ the ~~inspector~~ and ~~also his dog~~. ~~For extra powers you need coins and~~ with ~~their own help buy power up’s, new boards~~ and ~~additional. You can catch coins during this run OR you need to use this Subway Surfers Hack to come up~~ with it. ~~There~~ is ~~nothing simpler than get coins using this cheats tool~~. ~~Detailed instructions “How~~ to ~~have it? ” are below except for now I just give you basics~~.<br>~~First thing~~ is ~~getting~~ [~~http~~://~~Subwaysurferhackcheat~~.~~Neosites~~.~~com/subway-surfers-hack-cheat Subway Surfers Hack] from link below - you need to click “Download Now” button~~ and ~~you~~ will be ~~redirected to zippyshare~~.~~com~~. ~~Do you have it already? OK~~. ~~So lets get it started~~. ~~Run this cheats tool~~ and ~~plug your Android device~~ to ~~your computer~~. ~~You can use USB cable or bluetooth connection~~. ~~Now please detect your device~~ by ~~pressing~~ the ~~appropriate button~~. ~~After detecting device choose amount~~ of ~~resources~~ that ~~you need and click~~ the [~~https:~~//~~Www~~.~~Flickr.com~~/~~search~~/~~?q=magic+button magic button] called “Get it!”~~. ~~Now it~~ is ~~time for~~ the ~~best part~~ - ~~just wait~~ and ~~Coins~~ will be ~~added to your game account within~~ a ~~several seconds. The last thing~~ that ~~you must~~ to do is ~~restart game and unplug your device from computer.~~<br>~~Last question~~ is: ~~Who can use this particular Subway Surfers Hack and also cheats?~~. ~~There~~ is ~~an simple answer -~~ all ~~Android players throughout~~ the ~~world~~. ~~If you like playing this game~~ and ~~you have an Android system -~~ this ~~hack tool is~~ for ~~you~~ and ~~it should meet your needs exactly~~. ~~We had several distinct Android devices~~ (~~smartphones on~~ the ~~market~~ and ~~tablets~~) ~~including Sony Xperia Z, LG G2, Samsung Galaxy Tab 3, Nexus 7, Nexus 4, LG Flex, or brand new Samsung Universe S5~~ and [~~http~~://~~Www~~.~~Squidoo~~.~~com~~/~~search~~/~~results?q~~=~~Sony+Xperia Sony Xperia] Z2~~. ~~Subway Surfers Hack labored properly on those products~~. ~~So get it~~ and ~~check it yourself at this point~~. ~~It is 100% safe that you can~~.		In [[topology]], a '''coherent topology''' is one that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent with a family of subspaces if it is a '''topological union''' of those subspaces.

			==Definition==

			Let ''X'' be a [[topological space]] and let ''C'' = {''C''<sub>α</sub> : α ∈ ''A''} be a [[indexed family\|family]] of [[subspace (topology)\|subspace]]s of ''X'' (typically ''C'' will be a [[cover (topology)\|cover]] of ''X''). Then ''X'' is said to be '''coherent with ''C''''' (or '''determined by ''C''''')<ref>''X'' is also said to have the '''weak topology''' generated by ''C''. This is a potentially confusing name since the adjectives ''weak'' and ''strong'' are used with opposite meanings by different authors. In modern usage the term ''weak topology'' is synonymous with [[initial topology]] and ''strong topology'' is synonymous with [[final topology]]. It is the final topology that is being discussed here.</ref> if ''X'' has the [[final topology]] coinduced by the [[inclusion map]]s
			:<math>i_\alpha : C_\alpha \to X\qquad \alpha \in A.</math>
			By definition, this is the [[finest topology]] on ''X'' for which the inclusion maps are [[continuous function (topology)\|continuous]].

			Equivalently, ''X'' is coherent with ''C'' if either of the following two equivalent conditions holds:
			*A subset ''U'' is [[open set\|open]] in ''X'' if and only if ''U'' ∩ ''C''<sub>α</sub> is open in ''C''<sub>α</sub> for each α ∈ ''A''.
			*A subset ''U'' is [[closed set\|closed]] in ''X'' if and only if ''U'' ∩ ''C''<sub>α</sub> is closed in ''C''<sub>α</sub> for each α ∈ ''A''.

			Given a topological space ''X'' and any family of subspaces ''C'' there is unique topology on ''X'' which is coherent with ''C''. This topology will, in general, be [[finer topology\|finer]] than the given topology on ''X''.

			==Examples==

			*A topological space ''X'' is coherent with every [[open cover]] of ''X''.
			*A topological space ''X'' is coherent with every [[locally finite collection\|locally finite]] closed cover of ''X''.
			*A [[discrete space]] is coherent with every family of subspaces (including the [[empty set\|empty family]]).
			*A topological space ''X'' is coherent with a [[partition (set theory)\|partition]] of ''X'' if and only ''X'' is [[homeomorphic]] to the [[disjoint union (topology)\|disjoint union]] of the elements of the partition.
			*[[Finitely generated space]]s are those determined by the family of all [[finite topological space\|finite subspaces]].
			*[[Compactly generated space]]s are those determined by the family of all [[compact space\|compact subspace]]s.
			*A [[CW complex]] ''X'' is coherent with its family of ''n''-skeletons ''X''<sub>''n''</sub>.

			==Topological union==

			Let {''X''<sub>α</sub>} be a family of (not necessarily [[disjoint set\|disjoint]]) topological spaces such that the [[induced topology\|induced topologies]] agree on each [[intersection (set theory)\|intersection]] ''X''<sub>α</sub> ∩ ''X''<sub>β</sub>. Assume further that ''X''<sub>α</sub> ∩ ''X''<sub>β</sub> is closed in ''X''<sub>α</sub> for each α,β. Then the '''topological union''' of {''X''<sub>α</sub>} is the [[set-theoretic union]]
			:<math>X = \bigcup_{\alpha\in A}X_\alpha</math>
			together with the final topology coinduced by the inclusion maps <math>i_\alpha : X_\alpha \to X</math>. The inclusion maps will then be [[topological embedding]]s and ''X'' will be coherent with the subspaces {''X''<sub>α</sub>}.

			Conversely, if ''X'' is coherent with a family of subspaces {''C''<sub>α</sub>} that cover ''X'', then ''X'' is [[homeomorphic]] to the topological union of the family {''C''<sub>α</sub>}.

			One can form the topological union of an arbitrary family of topological spaces as above, but if the topologies do not agree on the intersections then the inclusions will not necessarily be embeddings.

			One can also describe the topological union by means of the [[disjoint union (topology)\|disjoint union]]. Specifically, if ''X'' is a topological union of the family {''X''<sub>α</sub>}, then ''X'' is homeomorphic to the [[quotient space\|quotient]] of the disjoint union of the family {''X''<sub>α</sub>} by the [[equivalence relation]]
			:<math>(x,\alpha) \sim (y,\beta) \Leftrightarrow x = y</math>
			for all α, β in ''A''. That is,
			:<math>X \cong \coprod_{\alpha\in A}X_\alpha / \sim.</math>

			If the spaces {''X''<sub>α</sub>} are all disjoint then the topological union is just the disjoint union.

			==Properties==

			Let ''X'' be coherent with a family of subspaces {''C''<sub>α</sub>}. A map ''f'' : ''X'' → ''Y'' is [[continuous function (topology)\|continuous]] if and only if the restrictions
			:<math>f\|_{C_\alpha} : C_\alpha \to Y\,</math>
			are continuous for each α ∈ ''A''. This [[universal property]] characterizes coherent topologies in the sense that a space ''X'' is coherent with ''C'' if and only if this property holds for all spaces ''Y'' and all functions ''f'' : ''X'' → ''Y''.

			Let ''X'' be determined by a [[cover (topology)\|cover]] ''C'' = {''C''<sub>α</sub>}. Then
			*If ''C'' is a [[refinement (topology)\|refinement]] of a cover ''D'', then ''X'' is determined by ''D''.
			*If ''D'' is a refinement of ''C'' and each ''C''<sub>α</sub> is determined by the family of all ''D''<sub>β</sub> contained in ''C''<sub>α</sub> then ''X'' is determined by ''D''.

			Let ''X'' be determined by {''C''<sub>α</sub>} and let ''Y'' be an open or closed [[subspace (topology)\|subspace]] of ''X''. Then ''Y'' is determined by {''Y'' ∩ ''C''<sub>α</sub>}.

			Let ''X'' be determined by {''C''<sub>α</sub>} and let ''f'' : ''X'' → ''Y'' be a [[quotient map]]. Then ''Y'' is determined by {f(''C''<sub>α</sub>)}.

			Let ''f'' : ''X'' → ''Y'' be a [[surjective map]] and suppose ''Y'' is determined by {''D''<sub>α</sub> : α ∈ ''A''}. For each α ∈ ''A'' let
			:<math>f_\alpha : f^{-1}(D_\alpha) \to D_\alpha\,</math>
			be the restriction of ''f'' to ''f''<sup>−1</sup>(''D''<sub>α</sub>). Then
			*If ''f'' is continuous and each ''f''<sub>α</sub> is a quotient map, then ''f'' is a quotient map.
			*''f'' is a [[closed map]] (resp. [[open map]]) if and only if each ''f''<sub>α</sub> is closed (resp. open).

			==Notes==

			<references/>

			==References==

			*{{cite encyclopedia \| last = Tanaka \| first = Yoshio \| editor = K.P. Hart, J. Nagata, and J.E. Vaughan \| title = Quotient Spaces and Decompositions \| encyclopedia = Encyclopedia of General Topology \| publisher = Elsevier Science \| location = Amsterdam \| year = 2004 \| pages = 43–46 \| isbn=0-444-50355-2}}
			*{{cite book \| last = Willard \| first = Stephen \| title = General Topology \| publisher = Addison-Wesley \| location = Reading, Massachusetts \| year = 1970 \| id = ISBN 0-486-43479-6 (Dover edition)}}

			[[Category:General topology]]

en>DeminJanu: added newport corps page on Optical Invariant as reference

2011-09-08T20:29:15Z

added newport corps page on Optical Invariant as reference

New page

We had 2 or 3 days off but we’re again now, for you, with the newest Subway People Hack! It was created for all players all over the world. Subway Surfers is a game which is called “endless running” game. In this game you're young hooligan who painted graffiti on the wall. For this reason you should escape from the inspector and also his dog. For extra powers you need coins and with their own help buy power up’s, new boards and additional. You can catch coins during this run OR you need to use this Subway Surfers Hack to come up with it. There is nothing simpler than get coins using this cheats tool. Detailed instructions “How to have it? ” are below except for now I just give you basics.<br>First thing is getting [http://Subwaysurferhackcheat.Neosites.com/subway-surfers-hack-cheat Subway Surfers Hack] from link below - you need to click “Download Now” button and you will be redirected to zippyshare.com. Do you have it already? OK. So lets get it started. Run this cheats tool and plug your Android device to your computer. You can use USB cable or bluetooth connection. Now please detect your device by pressing the appropriate button. After detecting device choose amount of resources that you need and click the [https://Www.Flickr.com/search/?q=magic+button magic button] called “Get it!”. Now it is time for the best part - just wait and Coins will be added to your game account within a several seconds. The last thing that you must to do is restart game and unplug your device from computer.<br>Last question is: Who can use this particular Subway Surfers Hack and also cheats?. There is an simple answer - all Android players throughout the world. If you like playing this game and you have an Android system - this hack tool is for you and it should meet your needs exactly. We had several distinct Android devices (smartphones on the market and tablets) including Sony Xperia Z, LG G2, Samsung Galaxy Tab 3, Nexus 7, Nexus 4, LG Flex, or brand new Samsung Universe S5 and [http://Www.Squidoo.com/search/results?q=Sony+Xperia Sony Xperia] Z2. Subway Surfers Hack labored properly on those products. So get it and check it yourself at this point. It is 100% safe that you can.

Lagrange invariant - Revision history

en>Dima373: Optics-stub

en>DeminJanu: added newport corps page on Optical Invariant as reference