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In [[mathematics]], more specifically [[topology]], a '''local homeomorphism''' is intuitively a [[Function (mathematics)|function]], ''f'', between [[topological space]]s that preserves local structure.
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==Formal definition==
Let ''X'' and ''Y'' be [[topological space]]s. A [[Function (mathematics)|function]] <math>f : X \to Y\,</math> is a '''local homeomorphism'''<ref>{{cite book | last=Munkres | first=James R. | authorlink=James Munkres | title=Topology | edition=2nd | publisher=[[Prentice Hall]] | year=2000 | isbn=0-13-181629-2}}</ref> if for every point ''x'' in ''X'' there exists an [[open set]] ''U'' containing ''x'', such that the [[image (mathematics)|image]] <math>f(U)</math> is open in ''Y'' and the [[restriction (mathematics)|restriction]] <math>f|_U : U\to f(U)\,</math> is a [[homeomorphism]].
 
==Examples==
By definition, every homeomorphism is also a '''local homeomorphism'''.
 
If ''U'' is an open subset of ''Y'' equipped with the [[subspace (topology)|subspace topology]], then the inclusion map ''i'' : ''U'' → ''Y'' is a local homeomorphism. Openness is essential here: the inclusion map of a non-open subset of ''Y'' never yields a local homeomorphism.
 
Every [[covering space|covering map]] is a local homeomorphism; in particular, the [[universal cover]] ''p'' : ''C'' &rarr; ''Y'' of a space ''Y'' is a local homeomorphism. In certain situations, the converse is true: if ''X'' and ''Y'' are [[locally compact space]]s and ''p'' : ''X'' → ''Y'' is a [[proper map|proper]] local homeomorphism, then ''p'' is a covering map.
 
Let ''f'' : ''S''<sup>1</sup> → ''S''<sup>1</sup> be the map that wraps the [[circle]] around itself ''n'' times (i.e. has [[winding number]] ''n''). This is a local homeomorphism for all non-zero ''n'', but a homeomorphism only in the cases where it is [[bijective]], i.e. ''n'' = 1 or -1.
 
It is shown in [[complex analysis]] that a complex [[holomorphic function|analytic]] function ''f'' gives a local homeomorphism precisely when the [[derivative]] ''f''&nbsp;&prime;(''z'') is non-zero for all ''z'' in the domain of ''f''. The function ''f''(''z'') = ''z''<sup>''n''</sup> on an open disk around 0 is not a local homeomorphism at 0 when ''n'' is at least 2. In that case 0 is a point of "[[ramification]]" (intuitively, ''n'' sheets come together there).
 
==Properties==
Every local homeomorphism is a [[continuous function (topology)|continuous]] and [[open map]]. A [[bijective]] local homeomorphism is therefore a homeomorphism.
 
A local homeomorphism  ''f'' : ''X'' → ''Y'' preserves "local" topological properties:
* ''X'' is [[locally connected space|locally connected]] if and only if ''f''(''X'') is
* ''X'' is [[locally connected space|locally path-connected]] if and only if ''f''(''X'') is
* ''X'' is [[locally compact]] if and only if ''f''(''X'') is
* ''X'' is [[first-countable space|first-countable]] if and only if ''f''(''X'') is
 
If ''f'' : ''X'' → ''Y'' is a local homeomorphism and ''U'' is an open subset of ''X'', then the restriction ''f''|<sub>''U''</sub> is also a local homeomorphism.
 
If ''f'' : ''X'' → ''Y'' and ''g'' : ''Y'' → ''Z'' are local homeomorphisms, then the composition ''gf'' : ''X'' → ''Z'' is also a local homeomorphism.
 
The local homeomorphisms with [[codomain]] ''Y'' stand in a natural 1-1 correspondence with the [[sheaf (mathematics)|sheaves]] of sets on ''Y''. Furthermore, every continuous map with codomain ''Y'' gives rise to a uniquely defined local homeomorphism with codomain ''Y'' in a natural way. All of this is explained in detail in the article on [[sheaf (mathematics)|sheaves]].
 
==See also==
*[[Local diffeomorphism]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Local Homeomorphism}}
[[Category:Homeomorphisms]]
[[Category:General topology]]

Latest revision as of 14:23, 5 May 2014

Were we inside a relationship with a guy with who you had ardentfervent feelings for and he broke up with you? Do we say to yourself all the time "I desire my spouse back"? If this is the case then read on to locate out what you are able to do to get ex back.

One other variable which will greatly affect your fat loss efforts is whether or not you may be properly measuring the caloric consumption. Even the most accurate bmr calculator in the planet cannot enable we in the event you are not monitoring the calories appropriately. There are many online calorie counting and food journaling websites to help you do this, however, certain difficult work on the piece is necessary. Be sure you may be weighing plus measuring a servings properly!

MyFitnessPal features a food diary, exercise log, online forums, plus blog with helpful fat reduction tips. It additionally delivers calculators for the heart rate, body mass index (BMI), basal metabolic rate (BMR), plus different nutritional values.

The suggested regime is three months and may extend to six months in some difficult instances. The oil is used by regulars too whom need to keep their figure and prevent accumulation of fat.

Next add the calories needed for a specific exercise we will be doing for that day to the bmr. For instance a female which weighs 165 pounds at 5'9", requirements 1650 calories for BMR (body weight X 10). Let's say she is moderately active thus add another 40% to her BMR: 1650 X .40 (40%) = 660 thus 1650 + 660 = 2310.

Needless to say, you'd not be eating much actual food if you ate high calorie foods like those, plus would usually feel hungry. You wish To feel full and happy. Choose foods which have lots of fiber, like fruits and entire grain breads, considering fiber makes we feel full. If you love pasta, try eating it with a light pesto rather of cream sauce. If the foods we eat are lower inside calories, you are able to eat more of them.

Finally, on the healthy eating plan, you need to let yourself one or 2 "cheats" per week. By doing this, you can talk oneself down from giving in whenever temptation suddenly arises. You could consult the food pyramid and the FDA's dietary guidelines plus never go below 500 calories less than the body requirements every day (unless a doctor provides we the ok to go lower than that).

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