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| {{distinguish|homomorphism}}
| | Most persons have this habit of doing all the stuff by themselves, irrespective of how critical or simple they are! These people won't let others interfere in their affairs. While this stance might function inside different regions of existence, it is actually really not how to reply when you need to fix the Windows registry. There are some jobs including removing spywares, virus and also obsolete registry entries, that are best left to specialist softwares. In this short article I may tell you why it really is critical to fix Windows registry NOW!<br><br>We need to understand several convenient and cheap ways that could resolve the problem of the computer and speed it up. The earlier we fix it, the less damage a computer gets. I will tell regarding several valuable techniques that usually assist you to speed up we computer.<br><br>It doesn't matter whether you may be not very clear regarding what rundll32.exe is. However remember that it plays an important character in keeping the stability of our computers plus the integrity of the program. Whenever certain software or hardware couldn't answer normally to the system procedure, comes the rundll32 exe error, that will be caused by corrupted files or lost information in registry. Usually, error content usually shows up at booting or the beginning of running a program.<br><br>The way to fix this issue is to first reinstall the program(s) causing the mistakes. There are a great deal of different programs that use this file, but 1 could have placed their own faulty variation of the file onto a program. By reinstalling any programs which are causing the error, you will not only let a PC to run the system correctly, however, a hot file might be placed onto your system - leaving the computer running as smoothly as possible again. If you try this, and discover it does not function, then we should look to update a program & any software you have on your PC. This will likely update the Msvcr71.dll file, allowing a computer to read it correctly again.<br><br>Whenever it comes to software, this is the vital piece because it is the 1 running the system because well as different programs needed inside a works. Always keep the cleanliness of your program from obsolete data by getting a good [http://bestregistrycleanerfix.com/regzooka regzooka]. Protect it from a virus online by providing a workable virus security system. You should equally have a monthly clean up by running your defragmenter system. This way it can enhance the performance of the computer plus for we to avoid any errors. If you think anything is incorrect with the software, and we don't know how to fix it then refer to a technician.<br><br>The initial thing you should do is to reinstall any program that shows the error. It's typical for many computers to have certain programs which require this DLL to show the error whenever we try and load it up. If you see a certain program show the error, you need to first uninstall that system, restart a PC and then resinstall the system again. This must replace the damaged ac1st16.dll file plus cure the error.<br><br>To accelerate your computer, you merely have to be able to get rid of all these junk files, allowing a computer to find just what it wants, whenever it wants. Luckily, there's a tool that enables us to do this easily plus immediately. It's a tool called a 'registry cleaner'.<br><br>What I would suggest is to search on your own for registry products. You are able to do this with a Google search. If you find goods, look for reviews and testimonials about the product. Next you are able to see how others like the product, plus how effectively it works. |
| {{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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Most persons have this habit of doing all the stuff by themselves, irrespective of how critical or simple they are! These people won't let others interfere in their affairs. While this stance might function inside different regions of existence, it is actually really not how to reply when you need to fix the Windows registry. There are some jobs including removing spywares, virus and also obsolete registry entries, that are best left to specialist softwares. In this short article I may tell you why it really is critical to fix Windows registry NOW!
We need to understand several convenient and cheap ways that could resolve the problem of the computer and speed it up. The earlier we fix it, the less damage a computer gets. I will tell regarding several valuable techniques that usually assist you to speed up we computer.
It doesn't matter whether you may be not very clear regarding what rundll32.exe is. However remember that it plays an important character in keeping the stability of our computers plus the integrity of the program. Whenever certain software or hardware couldn't answer normally to the system procedure, comes the rundll32 exe error, that will be caused by corrupted files or lost information in registry. Usually, error content usually shows up at booting or the beginning of running a program.
The way to fix this issue is to first reinstall the program(s) causing the mistakes. There are a great deal of different programs that use this file, but 1 could have placed their own faulty variation of the file onto a program. By reinstalling any programs which are causing the error, you will not only let a PC to run the system correctly, however, a hot file might be placed onto your system - leaving the computer running as smoothly as possible again. If you try this, and discover it does not function, then we should look to update a program & any software you have on your PC. This will likely update the Msvcr71.dll file, allowing a computer to read it correctly again.
Whenever it comes to software, this is the vital piece because it is the 1 running the system because well as different programs needed inside a works. Always keep the cleanliness of your program from obsolete data by getting a good regzooka. Protect it from a virus online by providing a workable virus security system. You should equally have a monthly clean up by running your defragmenter system. This way it can enhance the performance of the computer plus for we to avoid any errors. If you think anything is incorrect with the software, and we don't know how to fix it then refer to a technician.
The initial thing you should do is to reinstall any program that shows the error. It's typical for many computers to have certain programs which require this DLL to show the error whenever we try and load it up. If you see a certain program show the error, you need to first uninstall that system, restart a PC and then resinstall the system again. This must replace the damaged ac1st16.dll file plus cure the error.
To accelerate your computer, you merely have to be able to get rid of all these junk files, allowing a computer to find just what it wants, whenever it wants. Luckily, there's a tool that enables us to do this easily plus immediately. It's a tool called a 'registry cleaner'.
What I would suggest is to search on your own for registry products. You are able to do this with a Google search. If you find goods, look for reviews and testimonials about the product. Next you are able to see how others like the product, plus how effectively it works.