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| {{distinguish|homomorphism}}
| | Are you always having problems with a PC? Are we usually searching for methods to increase PC performance? Then this is the article you're trying to find. Here we will discuss a few of the most asked questions with regards to having you PC serve you well; how will I make my computer quicker for free? How to make my computer run faster?<br><br>Document files enable the consumer to input data, images, tables and additional elements to enhance the presentation. The only issue with this formatting compared to different file types such as .pdf for example is its ability to be commonly editable. This signifies which anyone viewing the file could change it by accident. Also, this file formatting is opened by additional programs but it does not guarantee which what we see inside the Microsoft Word application might nevertheless become the same when you view it using another system. However, it's still preferred by many computer users for its ease of utilize plus features.<br><br>Although this issue affects millions of computer users throughout the planet, there is an convenient method to fix it. You see, there's one reason for a slow loading computer, plus that's considering the PC cannot read the files it requires to run. In a nutshell, this simply means which when we do anything on Windows, it requirements to read up on how to do it. It's traditionally a surprisingly 'dumb' system, which has to have files to tell it to do everything.<br><br>Review your files and clean it up regularly. Destroy all unwanted and unused files because they only jam your computer program. It will definitely improve the speed of the computer and be careful which your computer do not infected by a virus. Remember constantly to update your antivirus software each time. If you do not use a computer truly often, you are able to take a free antivirus.<br><br>These are the results which the [http://bestregistrycleanerfix.com/registry-reviver registry reviver] found: 622 incorrect registry entries, 45,810 junk files, 15,643 unprotected privacy files, 8,462 bad Active X goods which were not blocked, 16 performance qualities which were not optimized, plus 4 updates that the computer needed.<br><br>The initial thing you need to do is to reinstall any system which shows the error. It's typical for many computers to have particular programs which need this DLL to show the error when you try and load it up. If you see a specific system show the error, you must initially uninstall that system, restart a PC and then resinstall the system again. This should replace the damaged ac1st16.dll file plus cure the error.<br><br>Google Chrome is my lifeline plus to the day fortunately. My all settings plus research connected bookmarks were saved inside Chrome plus stupidly I did not synchronize them with the Gmail to store them online. I couldn't afford to install new variation and sacrifice all my function settings. There was no method to retrieve the older settings. The just option left for me was to miraculously fix it browser in a way which all of the data and settings stored inside it are recovered.<br><br>There are numerous firms which offer the service of troubleshooting a PC every time you call them, all you need to do is signal up with them and for a small fee, you could have the machine always working perfectly plus serve we greater. |
| {{redirect|Topological equivalence|topological equivalence in dynamical systems|Topological conjugacy}}
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| [[Image:Mug and Torus morph.gif|thumb|right|A continuous deformation between a coffee [[mug]] and a [[torus|donut]] illustrating that they are homeomorphic. But there need not be a [[Homotopy|continuous deformation]] for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse.]]
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| In the [[mathematics|mathematical]] field of [[topology]], a '''homeomorphism''' or '''topological isomorphism''' or '''bicontinuous function''' is a [[continuous function]] between [[topological spaces]] that has a continuous [[inverse function]]. Homeomorphisms are the [[isomorphism]]s in the [[category of topological spaces]]—that is, they are the [[map (mathematics)|mappings]] that preserve all the [[topological property|topological properties]] of a given space. Two spaces with a homeomorphism between them are called '''homeomorphic''', and from a topological viewpoint they are the same. The word ''homeomorphism'' comes from the [[Greek language|Greek]] words ''[[wikt:ὅμοιος|ὅμοιος]]'' (''homoios'') = similar and ''[[wikt:μορφή|μορφή]]'' (''morphē'') = shape, form.
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| Roughly speaking, a topological space is a [[geometry|geometric]] object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a [[square (geometry)|square]] and a [[circle]] are homeomorphic to each other, but a [[sphere]] and a [[torus|donut]] are not. An often-repeated [[mathematical joke]] is that topologists can't tell their coffee cup from their donut,<ref>{{cite book|title=Differential Equations: A Dynamical Systems Approach. Part II: Higher-Dimensional Systems|first1=John H.|last1=Hubbard|first2=Beverly H.|last2=West|publisher=Springer|series=Texts in Applied Mathematics|volume=18|year=1995|isbn=978-0-387-94377-0|page=204|url=http://books.google.com/books?id=SHBj2oaSALoC&pg=PA204&dq=%22coffee+cup%22+topologist+joke#v=onepage&q=%22coffee%20cup%22%20topologist%20joke&f=false}}</ref> since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle.
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| Topology is the study of those properties of objects that do not change when homeomorphisms are applied. As [[Henri Poincaré]] famously said, [[mathematics]] is not the study of objects, but, instead, the relations (isomorphisms for instance) between them.<ref>{{cite book|last=Poincaré|first=Henri|authorlink=Henri Poincaré|title=Science and Hypothesis|chapter=Chapter II: Mathematical Magnitude and Experiment|url=https://en.wikisource.org/wiki/Science_and_Hypothesis/PART_I#b}}</ref>
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| ==Definition==
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| A [[function (mathematics)|function]] ''f'': ''X'' → ''Y'' between two [[topological space]]s (''X'', ''T<sub>X</sub>'') and (''Y'', ''T<sub>Y</sub>'') is called a '''homeomorphism''' if it has the following properties:
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| * ''f'' is a [[bijection]] ([[injective function|one-to-one]] and [[onto]]),
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| * ''f'' is [[Continuity (topology)|continuous]],
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| * the [[inverse function]] ''f''<sup> −1</sup> is continuous (f is an [[open mapping]]).
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| A function with these three properties is sometimes called '''bicontinuous'''. If such a function exists, we say ''X'' and ''Y'' are '''homeomorphic'''. A '''self-homeomorphism''' is a homeomorphism of a topological space and itself. The homeomorphisms form an [[equivalence relation]] on the [[class (set theory)|class]] of all topological spaces. The resulting [[equivalence class]]es are called '''homeomorphism classes'''.
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| ==Examples==
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| [[Image:Trefoil knot arb.png|thumb|right|240|A [[trefoil knot]] is homeomorphic to a circle, but not [[Homotopy#Isotopy|isotopic]]. Continuous mappings are not always realizable as deformations. Here the knot has been thickened to make the image understandable.]]
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| * The unit 2-[[ball (mathematics)|disc]] D<sup>2</sup> and the [[unit square]] in '''R'''<sup>2</sup> are homeomorphic.
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| * The open [[interval (mathematics)|interval]] (a, b) is homeomorphic to the [[real number]]s '''R''' for any a < b. (In this case, a bicontinuous forward mapping is given by {{math|''f'' {{=}} 1/(''x'' − ''a'') + 1/(''x'' − ''b'')}} while another such mapping is given by a scaled and translated version of the {{math|tan}} function).
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| * The [[product topology|product space]] [[Sphere|S<sup>1</sup>]] × S<sup>1</sup> and the two-[[dimension]]al [[torus]] are homeomorphic.
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| * Every [[uniform isomorphism]] and [[isometric isomorphism]] is a homeomorphism.
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| * The [[2-sphere]] with a single point removed is homeomorphic to the set of all points in '''R'''<sup>2</sup> (a 2-dimensional [[plane (mathematics)|plane]]).
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| * Let ''A'' be a commutative ring with unity and let ''S'' be a multiplicative subset of ''A''. Then Spec(''A''<sub>''S''</sub>) is homeomorphic to {{nowrap|1={''p'' ∈ Spec(''A'') : ''p'' ∩ ''S'' = ∅}.}}
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| * '''R'''<sup>''m''</sup> and '''R'''<sup>''n''</sup> are not homeomorphic for {{nowrap|1=''m'' ≠ ''n''.}}
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| * The Euclidean [[real line]] is not homeomorphic to the unit circle as a subspace of '''R'''<sup>''2''</sup> as the unit circle is compact as a subspace of Euclidean '''R'''<sup>''2''</sup> but the real line is not compact.
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| ==Notes==
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| The third requirement, that ''f''<sup> −1</sup> be continuous, is essential. Consider for instance the function ''f'': <nowiki>[0, 2π)</nowiki> → S<sup>1</sup> defined by ''f''(φ) = (cos(φ), sin(φ)). This function is bijective and continuous, but not a homeomorphism (S<sup>1</sup> is compact but <nowiki>[0, 2π)</nowiki> is not).
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| Homeomorphisms are the [[isomorphism]]s in the [[category of topological spaces]]. As such, the composition of two homeomorphisms is again a homeomorphism, and the set of all self-homeomorphisms ''X'' → ''X'' forms a [[group (mathematics)|group]], called the '''[[homeomorphism group]]''' of ''X'', often denoted Homeo(''X''); this group can be given a topology, such as the [[compact-open topology]], making it a [[topological group]].
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| For some purposes, the homeomorphism group happens to be too big, but
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| by means of the [[Homotopy#Isotopy|isotopy]] relation, one can reduce this group to the
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| [[mapping class group]].
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| Similarly, as usual in category theory, given two spaces that are homeomorphic, the space of homeomorphisms between them, Homeo(''X,'' ''Y''), is a [[torsor]] for the homeomorphism groups Homeo(''X'') and Homeo(''Y''), and given a specific homeomorphism between ''X'' and ''Y'', all three sets are identified.
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| ==Properties==
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| * Two homeomorphic spaces share the same [[topological property|topological properties]]. For example, if one of them is [[compact space|compact]], then the other is as well; if one of them is [[connectedness|connected]], then the other is as well; if one of them is [[Hausdorff space|Hausdorff]], then the other is as well; their [[homotopy]] & [[homology group]]s will coincide. Note however that this does not extend to properties defined via a [[metric space|metric]]; there are metric spaces that are homeomorphic even though one of them is [[completeness (topology)|complete]] and the other is not.
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| * A homeomorphism is simultaneously an [[open mapping]] and a [[closed mapping]]; that is, it maps [[open set]]s to open sets and [[closed set]]s to closed sets.
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| * Every self-homeomorphism in <math>S^1</math> can be extended to a self-homeomorphism of the whole disk <math>D^2</math> ([[Alexander's trick]]).
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| ==Informal discussion==
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| The intuitive criterion of stretching, bending, cutting and gluing back together takes a certain amount of practice to apply correctly—it may not be obvious from the description above that deforming a [[line segment]] to a point is impermissible, for instance. It is thus important to realize that it is the formal definition given above that counts.
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| This characterization of a homeomorphism often leads to confusion with the concept of [[homotopy]], which is actually ''defined'' as a continuous deformation, but from one ''function'' to another, rather than one space to another. In the case of a homeomorphism, envisioning a continuous deformation is a mental tool for keeping track of which points on space ''X'' correspond to which points on ''Y''—one just follows them as ''X'' deforms. In the case of homotopy, the continuous deformation from one map to the other is of the essence, and it is also less restrictive, since none of the maps involved need to be one-to-one or onto. Homotopy does lead to a relation on spaces: [[homotopy equivalence]].
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| There is a name for the kind of deformation involved in visualizing a homeomorphism. It is (except when cutting and regluing are required) an [[homotopy|isotopy]] between the [[identity function|identity map]] on ''X'' and the homeomorphism from ''X'' to ''Y''. | |
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| ==See also==
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| *[[Local homeomorphism]]
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| *[[Diffeomorphism]]
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| *[[Uniform isomorphism]] is an isomorphism between [[uniform spaces]]
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| *[[Isometric isomorphism]] is an isomorphism between [[metric spaces]]
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| *[[Dehn twist]]
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| *[[Homeomorphism (graph theory)]] (closely related to graph subdivision)
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| *[[Homotopy#Isotopy]]
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| *[[Mapping class group]]
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| *[[Poincaré conjecture]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{springer|title=Homeomorphism|id=p/h047600}}
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| *{{planetmath reference|id=912|title=Homeomorphism}}
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| [[Category:Homeomorphisms| ]]
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| [[Category:Functions and mappings]]
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Are you always having problems with a PC? Are we usually searching for methods to increase PC performance? Then this is the article you're trying to find. Here we will discuss a few of the most asked questions with regards to having you PC serve you well; how will I make my computer quicker for free? How to make my computer run faster?
Document files enable the consumer to input data, images, tables and additional elements to enhance the presentation. The only issue with this formatting compared to different file types such as .pdf for example is its ability to be commonly editable. This signifies which anyone viewing the file could change it by accident. Also, this file formatting is opened by additional programs but it does not guarantee which what we see inside the Microsoft Word application might nevertheless become the same when you view it using another system. However, it's still preferred by many computer users for its ease of utilize plus features.
Although this issue affects millions of computer users throughout the planet, there is an convenient method to fix it. You see, there's one reason for a slow loading computer, plus that's considering the PC cannot read the files it requires to run. In a nutshell, this simply means which when we do anything on Windows, it requirements to read up on how to do it. It's traditionally a surprisingly 'dumb' system, which has to have files to tell it to do everything.
Review your files and clean it up regularly. Destroy all unwanted and unused files because they only jam your computer program. It will definitely improve the speed of the computer and be careful which your computer do not infected by a virus. Remember constantly to update your antivirus software each time. If you do not use a computer truly often, you are able to take a free antivirus.
These are the results which the registry reviver found: 622 incorrect registry entries, 45,810 junk files, 15,643 unprotected privacy files, 8,462 bad Active X goods which were not blocked, 16 performance qualities which were not optimized, plus 4 updates that the computer needed.
The initial thing you need to do is to reinstall any system which shows the error. It's typical for many computers to have particular programs which need this DLL to show the error when you try and load it up. If you see a specific system show the error, you must initially uninstall that system, restart a PC and then resinstall the system again. This should replace the damaged ac1st16.dll file plus cure the error.
Google Chrome is my lifeline plus to the day fortunately. My all settings plus research connected bookmarks were saved inside Chrome plus stupidly I did not synchronize them with the Gmail to store them online. I couldn't afford to install new variation and sacrifice all my function settings. There was no method to retrieve the older settings. The just option left for me was to miraculously fix it browser in a way which all of the data and settings stored inside it are recovered.
There are numerous firms which offer the service of troubleshooting a PC every time you call them, all you need to do is signal up with them and for a small fee, you could have the machine always working perfectly plus serve we greater.