|
|
| Line 1: |
Line 1: |
| {{distinguish|Separated space}}
| | The second step is to talk to your physician. Before, you make any changes in your current weight-reduction plan, it's good to inform your doctor. The doctor will give [http://quality-ebooks.com/04Q lose weight airdyne] you even more helpful tips on making eating regimen and way of life modifications, that may additional guide you in a healthy weight loss.<br><br>There are many online weight reduction programs and one of the crucial fashionable ones is Weight Watchers. Not like other diets, in this on-line weight reduction program, there are no food or meals groups that you're forbidden to eat. Every item of meals that you eat, has a degree assigned to it. This point assignment is predicated on the kind of meals, energy it contains and its portion dimension. If you be part of Weight Watchers weight reduction program, you have to hold within a specified number of points in a day that's personalized just for you. You must strictly preserve a track of what you are eating.<br><br>Weight Watchers is an effective weight loss method, which has helped millions of individuals drop pounds and maintain it, through the years. Such is the recognition of Weight Watchers that eating places like Applebees have started giving out Weight Watchers eating out factors for his or her menu. Nevertheless, not every restaurant that you simply visit has a special Weight Watchers points information for fellow diners. This may drive the dieters to go off their weight loss plan plan each time they eat out. Considering that an average American eats out for not less than 4 instances a week, exceeding their level limit can show to be very costly for dieters.<br><br>These days people are so well being and health aware that they don't hesitate to attempt alternative ways to drop some weight. Strenuous train regimes and weight reduction diets are the 2 things that come to our mind once we wish to reduce [http://quality-ebooks.com/04Q lose weight fast aerobic exercise] weight fast. However as it's mentioned to discover a remedy for any well being ailment we have to know its underlying trigger, likewise for weight achieve or obesity poor eating habits or eating foods which might be excessive in calories and fats however low in nutrients are the foremost contributing elements.<br><br>This is yet one more means which you can use to follow Weight Watchers free of charge. Weight Watchers holds many monthly meetings, and some of them are free of value so that increasingly weight reduction aspirants can attend them. The Weight Watchers operators not only hold free conferences, but additionally they reward you for those who decide to affix them or achieve weight reduction with their methods. These meetings are principally free for first timers too. Attending the conferences provides you detailed details about the Weight Watchers weight loss program. Attending free meetings lets you resolve whether you actually wish to follow Weight Watchers. |
| | |
| In [[mathematics]] a [[topological space]] is called '''separable''' if it contains a [[countable set|countable]], [[dense (topology)|dense]] subset; that is, there exists a [[sequence]] <math>\{ x_n \}_{n=1}^{\infty} </math> of elements of the space such that every nonempty [[open subset]] of the space contains at least one element of the sequence.
| |
| | |
| Like the other [[axioms of countability]], separability is a "limitation on size", not necessarily in terms of [[cardinality]] (though, in the presence of the [[Hausdorff space|Hausdorff axiom]], this does turn out to be the case; see below) but in a more subtle topological sense. In particular, every [[continuous function]] on a separable space whose image is a subset of a Hausdorff space is determined by its values on the countable dense subset.
| |
| | |
| Contrast separability with the related notion of [[second countability]], which is in general stronger but equivalent on the class of [[metrizable]] spaces.
| |
| | |
| ==First examples==
| |
| Any topological space which is itself [[finite set|finite]] or [[countably infinite]] is separable, for the whole space is a countable dense subset of itself. An important example of an uncountable separable space is the [[real line]], in which the [[rational numbers]] form a countable dense subset. Similarly the set of all vectors <math>(r_1,\ldots,r_n) \in \mathbb{R}^n</math> in which <math>r_i</math> is rational for all ''i'' is a countable dense subset of <math>\mathbb{R}^n</math>; so for every <math>n</math> the <math>n</math>-dimensional [[Euclidean space]] is separable.
| |
| | |
| A simple example of a space which is not separable is a [[discrete space]] of uncountable cardinality.
| |
| | |
| Further examples are given below.
| |
| | |
| ==Separability versus second countability==
| |
| Any [[second-countable space]] is separable: if <math>\scriptstyle \{U_n\}</math> is a countable base, choosing any <math>\scriptstyle x_n \in U_n</math> from the non-empty <math>\scriptstyle U_n</math> gives a countable dense subset. Conversely, a [[metrizable space]] is separable if and only if it is second countable, which is the case if and only if it is [[Lindelöf space|Lindelöf]].
| |
| | |
| To further compare these two properties:
| |
| * An arbitrary [[subspace (topology)|subspace]] of a second countable space is second countable; subspaces of separable spaces need not be separable (see below).
| |
| * Any continuous image of a separable space is separable {{harv|Willard|1970|loc=Th. 16.4a}}.; even a [[quotient topology|quotient]] of a second countable space need not be second countable.
| |
| * A [[product topology|product]] of at most continuum many separable spaces is separable. A countable product of second countable spaces is second countable, but an uncountable product of second countable spaces need not even be first countable.
| |
| | |
| ==Cardinality==
| |
| The property of separability does not in and of itself give any limitations on the [[cardinality]] of a topological space: any set endowed with the [[trivial topology]] is separable, as well as second countable, [[quasi-compact]], and [[connected space|connected]]. The "trouble" with the trivial topology is its poor separation properties: its [[Kolmogorov quotient]] is the one-point space.
| |
| | |
| A [[first countable]], separable Hausdorff space (in particular, a separable metric space) has at most the [[cardinality of the continuum|continuum cardinality]] ''c''. In such a space, [[closure (topology)|closure]] is determined by limits of sequences and any sequence has at most one limit, so there is a surjective map from the set of convergent sequences with values in the countable dense subset to the points of X.
| |
| | |
| A separable Hausdorff space has cardinality at most <math>2^c</math>, where ''c'' is the cardinality of the continuum. For this closure is characterized in terms of limits of [[filter (mathematics)#Filters in topology|filter bases]]: if ''Y'' is a subset of ''X'' and ''z'' is a point of ''X'', then ''z'' is in the closure of ''Y'' if and only if there exists a filter base ''B'' consisting of subsets of ''Y'' which converges to ''z''. The cardinality of the set <math>S(Y)</math> of such filter bases is at most <math>2^{2^{|Y|}}</math>. Moreover, in a Hausdorff space, there is at most one limit to every filter base. Therefore, there is a surjection <math>S(Y) \rightarrow X</math> when <math>\bar Y=X.</math>
| |
| | |
| The same arguments establish a more general result: suppose that a Hausdorff topological space ''X'' contains a dense subset of cardinality <math>\kappa</math>.
| |
| Then ''X'' has cardinality at most <math>2^{2^{\kappa}}</math> and cardinality at most <math>2^{\kappa}</math> if it is first countable.
| |
| | |
| The product of at most continuum many separable spaces is a separable space {{harv | Willard | 1970 | loc=Th 16.4c | p=109 }}. In particular the space <math>\mathbb{R}^{\mathbb{R}}</math> of all functions from the real line to itself, endowed with the product topology, is a separable Hausdorff space of cardinality <math>2^c</math>. More generally, if κ is any infinite cardinal, then a product of at most 2<sup>κ</sup> spaces with dense subsets of size at most κ has itself a dense subset of size at most κ ([[Hewitt–Marczewski–Pondiczery theorem]]).
| |
| | |
| ==Constructive mathematics==
| |
| Separability is especially important in [[numerical analysis]] and [[Mathematical constructivism|constructive mathematics]], since many theorems that can be proved for nonseparable spaces have constructive proofs only for separable spaces. Such constructive proofs can be turned into [[algorithm]]s for use in numerical analysis, and they are the only sorts of proofs acceptable in constructive analysis. A famous example of a theorem of this sort is the [[Hahn–Banach theorem]].
| |
| | |
| ==Further examples==
| |
| ===Separable spaces===
| |
| * Every compact [[metric space]] (or metrizable space) is separable.
| |
| * The space of all continuous functions from a [[Compact space|compact]] subset of '''R'''<sup>n</sup> into '''R''' is separable.
| |
| * The [[Lp space|Lebesgue spaces]] '''L'''<sup>p</sup>, over a [[separable measure space]], are separable for any 1 ≤ ''p'' < ∞.
| |
| * Any topological space which is the union of a countable number of separable subspaces is separable. Together, these first two examples give a different proof that ''n''-dimensional Euclidean space is separable.
| |
| * It follows from the [[Stone-Weierstrass theorem|Weierstrass approximation theorem]] that the set '''Q'''[t] of polynomials with rational coefficients is a countable dense subset of the space C([0,1]) of [[continuous function]]s on the [[unit interval]] [0,1] with the metric of [[uniform convergence]]. The [[Banach-Mazur theorem]] asserts that any separable [[Banach space]] is isometrically isomorphic to a closed [[linear subspace]] of C([0,1]).
| |
| * A [[Hilbert space]] is separable if and only if it has a countable [[orthonormal basis]], it follows that any separable, infinite-dimensional Hilbert space is isometric to ℓ<sup>2</sup>.
| |
| * An example of a separable space that is not second-countable is '''R'''<sub>llt</sub>, the set of real numbers equipped with the [[lower limit topology]].
| |
| | |
| ===Non-separable spaces===
| |
| * The [[first uncountable ordinal|first uncountable ordinal ω<sub>1</sub>]] in its [[order topology]] is not separable.
| |
| * The [[Banach space]] ''l''<sup>∞</sup> of all bounded real sequences, with the [[uniform norm|supremum norm]], is not separable. The same holds for '''L'''<sup>∞</sup>.
| |
| * The [[Banach space]] of [[Bounded variation|functions of bounded variation]] is not separable; note however that this space has very important applications in [[mathematics]], [[physics]] and [[engineering]].
| |
| | |
| ==Properties==
| |
| * A [[subspace (topology)|subspace]] of a separable space need not be separable (see the [[Sorgenfrey plane]] and the [[Moore plane]]), but every ''open'' subspace of a separable space is separable, {{harv|Willard|1970|loc=Th 16.4b}}. Also every subspace of a separable [[metric space]] is separable.
| |
| * In fact, every topological space is a subspace of a separable space of the same [[cardinality]]. A construction adding at most countably many points is given in {{harv|Sierpinski|1952|p=49}}.
| |
| * The set of all real-valued continuous functions on a separable space has a cardinality less than or equal to ''c''. This follows since such functions are determined by their values on dense subsets.
| |
| * From the above property, one can deduce the following: If ''X'' is a separable space having an uncountable closed discrete subspace, then ''X'' cannot be [[normal space|normal]]. This shows that the [[Sorgenfrey plane]] is not normal.
| |
| *For a [[compact space|compact]] [[Hausdorff space|Hausdorff]] space ''X'', the following are equivalent:
| |
| ::(i) ''X'' is second countable.
| |
| ::(ii) The space <math>\mathcal{C}(X,\mathbb{R})</math> of continuous real-valued functions on ''X'' with the [[uniform norm|supremum norm]] is separable.
| |
| ::(iii) ''X'' is metrizable.
| |
| | |
| ===Embedding separable metric spaces===
| |
| * Every separable metric space is [[homeomorphic]] to a subset of the [[Hilbert cube]]. This is established in the proof of the [[Urysohn metrization theorem]].
| |
| * Every separable metric space is [[Isometry|isometric]] to a subset of the (non-separable) [[Banach space]] ''l''<sup>∞</sup> of all bounded real sequences with the [[uniform norm|supremum norm]]; this is known as the Fréchet embedding. {{harv | Heinonen | 2003}}
| |
| * Every separable metric space is isometric to a subset of C([0,1]), the separable Banach space of continuous functions [0,1]→'''R''', with the [[uniform norm|supremum norm]]. This is due to [[Stefan Banach]]. {{harv | Heinonen | 2003}}
| |
| * Every separable metric space is isometric to a subset of the [[Urysohn universal space]].
| |
| | |
| ==References==
| |
| *{{Citation | last1=Kelley | first1=John L. | author1-link=John L. Kelley | title=General Topology | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90125-1 | mr=0370454 | year=1975}}
| |
| *{{Citation | last1=Sierpiński | first1=Wacław | author1-link=Wacław Sierpiński | title=General topology | publisher=University of Toronto Press | location=Toronto, Ont. | series=Mathematical Expositions, No. 7 | mr=0050870 | year=1952}}
| |
| *{{Citation | last1=Steen | first1=Lynn Arthur | author1-link=Lynn Arthur Steen | last2=Seebach | first2=J. Arthur Jr. | author2-link=J. Arthur Seebach, Jr. | title=[[Counterexamples in Topology]] | origyear=1978 | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=[[Dover Publications|Dover]] reprint of 1978 | isbn=978-0-486-68735-3 | mr=507446 | year=1995}}
| |
| *{{Citation | last1=Willard | first1=Stephen | title=General Topology | publisher=[[Addison-Wesley]] | isbn=978-0-201-08707-9 | mr=0264581 | year=1970}}
| |
| *{{Citation|title=Geometric embeddings of metric spaces|url=http://www.math.jyu.fi/research/reports/rep90.pdf|author=Juha Heinonen|date=January 2003|accessdate=6 February 2009}}
| |
| {{Use dmy dates|date=September 2010}}
| |
| | |
| {{DEFAULTSORT:Separable Space}}
| |
| [[Category:General topology]]
| |
| [[Category:Properties of topological spaces]]
| |
The second step is to talk to your physician. Before, you make any changes in your current weight-reduction plan, it's good to inform your doctor. The doctor will give lose weight airdyne you even more helpful tips on making eating regimen and way of life modifications, that may additional guide you in a healthy weight loss.
There are many online weight reduction programs and one of the crucial fashionable ones is Weight Watchers. Not like other diets, in this on-line weight reduction program, there are no food or meals groups that you're forbidden to eat. Every item of meals that you eat, has a degree assigned to it. This point assignment is predicated on the kind of meals, energy it contains and its portion dimension. If you be part of Weight Watchers weight reduction program, you have to hold within a specified number of points in a day that's personalized just for you. You must strictly preserve a track of what you are eating.
Weight Watchers is an effective weight loss method, which has helped millions of individuals drop pounds and maintain it, through the years. Such is the recognition of Weight Watchers that eating places like Applebees have started giving out Weight Watchers eating out factors for his or her menu. Nevertheless, not every restaurant that you simply visit has a special Weight Watchers points information for fellow diners. This may drive the dieters to go off their weight loss plan plan each time they eat out. Considering that an average American eats out for not less than 4 instances a week, exceeding their level limit can show to be very costly for dieters.
These days people are so well being and health aware that they don't hesitate to attempt alternative ways to drop some weight. Strenuous train regimes and weight reduction diets are the 2 things that come to our mind once we wish to reduce lose weight fast aerobic exercise weight fast. However as it's mentioned to discover a remedy for any well being ailment we have to know its underlying trigger, likewise for weight achieve or obesity poor eating habits or eating foods which might be excessive in calories and fats however low in nutrients are the foremost contributing elements.
This is yet one more means which you can use to follow Weight Watchers free of charge. Weight Watchers holds many monthly meetings, and some of them are free of value so that increasingly weight reduction aspirants can attend them. The Weight Watchers operators not only hold free conferences, but additionally they reward you for those who decide to affix them or achieve weight reduction with their methods. These meetings are principally free for first timers too. Attending the conferences provides you detailed details about the Weight Watchers weight loss program. Attending free meetings lets you resolve whether you actually wish to follow Weight Watchers.