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| In [[topology]] and related areas of [[mathematics]], a '''product space''' is the [[cartesian product]] of a family of [[topological space]]s equipped with a [[natural topology]] called the '''product topology'''. This topology differs from another, perhaps more obvious, topology called the [[box topology]], which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a [[product (category theory)|categorical product]] of its factors, whereas the box topology is [[Comparison of topologies|too fine]]; this is the sense in which the product topology is "natural".
| | A great deal of people inside the antivirus industry were afraid that Microsoft was going to kill the antivirus marketplace. This hasn't occurred yet, even though Microsoft is operating OK inside independent antivirus tests.<br><br>These are certain tiny points about why Norton is the most effective answer for we. There are several additional reasons that I recommend Norton Anti Virus, and for a full right-up please visit my website. You can follow a link to the page talking particularly about Norton at the bottom of this post beneath Appendix A.<br><br>Like its Windows counterpart, BitDefender [http://leadingpcsoftware.com/best-antivirus-software/ antivirus software reviews] best software for Mac is strong, flexible and user friendly. It is totally capable of shielding your Mac OS X from all types of malware threats.<br><br>Although the free software is good, the maximum amount of real time security comes with the paid, specialist versions. That is the method the software businesses shape the stuff. The superior news? Most of the paid versions are truly not which expensive.<br><br>Click START, than CONTROL PANEL. 2.When the Control Panel opens, click USER ACCOUNTS. 3.Click CHANGE AN ACCOUNT. four.Click on the name of the account we want to remove. 5.Click REMOVE THIS ACCOUNT. 6.When provided the choice to "delete files" or "keep files", choose to cut them. 7.After removing all unused User Accounts, restart the computer. Windows Updates Make sure we have all newest Windows Updates installed by visiting update.microsoft.com and following the instructions found on the website.<br><br>11. After 30 seconds 5 minutes you will be loaded into windows safe mode (it claims safe mode on all 4 corners of the screen). Windows may ask should you if you would like to enter secure mode. Choose Yes.<br><br>For more information about AVG anti-virus software, feel free to pay a visit to the AVG free anti-virus download home page. Make sure your computer is compatible before ever creating changes with it. Many protection programs and software are not compatible with each alternative, thus be sure and just create changes to your computer when you know what you may be doing. If you do not understand what we are doing I would advise having a computer technician create the changes for you. |
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| ==Definition==
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| Given ''X'' such that
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| :<math>X := \prod_{i \in I} X_i,</math>
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| is the Cartesian product of the topological spaces ''X<sub>i</sub>'', [[index set|indexed]] by <math>i \in I</math>, and the '''[[projection (set theory)|canonical projections]]''' ''p<sub>i</sub>'' : ''X'' → ''X<sub>i</sub>'', the '''product topology''' on ''X'' is defined to be the [[coarsest topology]] (i.e. the topology with the fewest open sets) for which all the projections ''p<sub>i</sub>'' are [[continuous (topology)|continuous]]. The product topology is sometimes called the '''Tychonoff topology'''.
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| The open sets in the product topology are unions (finite or infinite) of sets of the form <math>\prod_{i\in I} U_i</math>, where each ''U<sub>i</sub>'' is open in ''X<sub>i</sub>'' and ''U''<sub>''i''</sub> ≠ ''X''<sub>''i''</sub> for only finitely many ''i''. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the ''X<sub>i</sub>'' gives a basis for the product <math>\prod_{i\in I} X_i</math>.
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| The product topology on ''X'' is the topology generated by sets of the form ''p<sub>i</sub>''<sup>−1</sup>(''U''), where ''i'' is in ''I '' and ''U'' is an open subset of ''X<sub>i</sub>''. In other words, the sets {''p<sub>i</sub>''<sup>−1</sup>(''U'')} form a [[subbase]] for the topology on ''X''. A [[subset]] of ''X'' is open if and only if it is a (possibly infinite) [[union (set theory)|union]] of [[intersection (set theory)|intersections]] of finitely many sets of the form ''p<sub>i</sub>''<sup>−1</sup>(''U''). The ''p<sub>i</sub>''<sup>−1</sup>(''U'') are sometimes called [[open cylinder]]s, and their intersections are [[cylinder set]]s.
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| In general, the product of the topologies of each ''X<sub>i</sub>'' forms a basis for what is called the [[box topology]] on ''X''. In general, the box topology is [[finer topology|finer]] than the product topology, but for finite products they coincide.
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| == Examples ==
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| If one starts with the [[standard topology]] on the [[real line]] '''R''' and defines a topology on the product of ''n'' copies of '''R''' in this fashion, one obtains the ordinary [[Euclidean topology]] on '''R'''<sup>''n''</sup>.
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| The [[Cantor set]] is [[homeomorphic]] to the product of [[countable|countably many]] copies of the [[discrete space]] {0,1} and the space of [[irrational number]]s is homeomorphic to the product of countably many copies of the [[natural number]]s, where again each copy carries the discrete topology. | |
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| Several additional examples are given in the article on the [[initial topology]].
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| == Properties ==
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| The product space ''X'', together with the canonical projections, can be characterized by the following [[universal property]]: If ''Y'' is a topological space, and for every ''i'' in ''I'', ''f<sub>i</sub>'' : ''Y'' → ''X<sub>i</sub>'' is a continuous map, then there exists ''precisely one'' continuous map ''f'' : ''Y'' → ''X'' such that for each ''i'' in ''I'' the following diagram [[commutative diagram|commutes]]:
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| [[Image:CategoricalProduct-02.png|center|Characteristic property of product spaces]]
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| This shows that the product space is a [[product (category theory)|product]] in the [[category of topological spaces]]. It follows from the above universal property that a map ''f'' : ''Y'' → ''X'' is continuous [[if and only if]] ''f<sub>i</sub>'' = ''p<sub>i</sub>'' o ''f'' is continuous for all ''i'' in ''I''. In many cases it is easier to check that the component functions ''f<sub>i</sub>'' are continuous. Checking whether a map ''g'' : ''X''→ ''Z'' is continuous is usually more difficult; one tries to use the fact that the ''p<sub>i</sub>'' are continuous in some way.
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| In addition to being continuous, the canonical projections ''p<sub>i</sub>'' : ''X'' → ''X<sub>i</sub>'' are [[open map]]s. This means that any open subset of the product space remains open when projected down to the ''X<sub>i</sub>''. The converse is not true: if ''W'' is a [[subspace (topology)|subspace]] of the product space whose projections down to all the ''X<sub>i</sub>'' are open, then ''W'' need not be open in ''X''. (Consider for instance ''W'' = '''R'''<sup>2</sup> \ (0,1)<sup>2</sup>.) The canonical projections are not generally [[closed map]]s (consider for example the closed set <math>\{(x,y) \in \mathbb{R}^2 \mid xy = 1\},</math> whose projections onto both axes are '''R''' \ {0}).
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| The product topology is also called the ''topology of pointwise convergence'' because of the following fact: a [[sequence]] (or [[Net (mathematics)|net]]) in ''X'' converges if and only if all its projections to the spaces ''X''<sub>''i''</sub> converge. In particular, if one considers the space ''X'' = '''R'''<sup>''I''</sup> of all [[real number|real]] valued [[function (mathematics)|function]]s on ''I'', convergence in the product topology is the same as pointwise convergence of functions.
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| Any product of closed subsets of ''X<sub>i</sub>'' is a closed set in ''X''.
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| An important theorem about the product topology is [[Tychonoff's theorem]]: any product of [[compact space]]s is compact. This is easy to show for finite products, while the general statement is equivalent to the [[axiom of choice]].
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| == Relation to other topological notions ==
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| * Separation
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| ** Every product of [[T0 space|T<sub>0</sub> space]]s is T<sub>0</sub>
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| ** Every product of [[T1 space|T<sub>1</sub> space]]s is T<sub>1</sub>
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| ** Every product of [[Hausdorff space]]s is Hausdorff<ref>{{planetmath reference|id=4317|title=Product topology preserves the Hausdorff property}}</ref>
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| ** Every product of [[regular space]]s is regular
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| ** Every product of [[Tychonoff space]]s is Tychonoff
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| ** A product of [[normal space]]s ''need not'' be normal
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| * Compactness
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| ** Every product of compact spaces is compact ([[Tychonoff's theorem]])
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| ** A product of [[locally compact space]]s ''need not'' be locally compact. However, an arbitrary product of locally compact spaces where all but finitely many are compact ''is'' locally compact (This condition is sufficient and necessary).
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| * Connectedness
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| ** Every product of [[connectedness|connected]] (resp. path-connected) spaces is connected (resp. path-connected)
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| ** Every product of hereditarily disconnected spaces is hereditarily disconnected.
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| ==Axiom of choice==
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| The [[axiom of choice]] is equivalent to the statement that the product of a collection of non-empty sets is non-empty. The proof is easy enough: one needs only to pick an element from each set to find a representative in the product. Conversely, a representative of the product is a set which contains exactly one element from each component.
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| The axiom of choice occurs again in the study of (topological) product spaces; for example, [[Tychonoff's theorem]] on compact sets is a more complex and subtle example of a statement that is equivalent to the axiom of choice.
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| == See also ==
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| *[[Disjoint union (topology)]]
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| *[[Initial topology|Projective limit topology]]
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| *[[Quotient space]]
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| *[[Subspace (topology)]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book |last=Willard |first=Stephen |title=General Topology |year=1970 |publisher=Addison-Wesley Pub. Co. |location=Reading, Mass. |isbn=0486434796 |url=http://store.doverpublications.com/0486434796.html |accessdate=13 February 2013}}
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| ==External links==
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| * {{planetmath reference|id=3100|title=product topology}}
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| [[Category:Topology]]
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| [[Category:General topology]]
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| [[Category:Binary operations]]
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A great deal of people inside the antivirus industry were afraid that Microsoft was going to kill the antivirus marketplace. This hasn't occurred yet, even though Microsoft is operating OK inside independent antivirus tests.
These are certain tiny points about why Norton is the most effective answer for we. There are several additional reasons that I recommend Norton Anti Virus, and for a full right-up please visit my website. You can follow a link to the page talking particularly about Norton at the bottom of this post beneath Appendix A.
Like its Windows counterpart, BitDefender antivirus software reviews best software for Mac is strong, flexible and user friendly. It is totally capable of shielding your Mac OS X from all types of malware threats.
Although the free software is good, the maximum amount of real time security comes with the paid, specialist versions. That is the method the software businesses shape the stuff. The superior news? Most of the paid versions are truly not which expensive.
Click START, than CONTROL PANEL. 2.When the Control Panel opens, click USER ACCOUNTS. 3.Click CHANGE AN ACCOUNT. four.Click on the name of the account we want to remove. 5.Click REMOVE THIS ACCOUNT. 6.When provided the choice to "delete files" or "keep files", choose to cut them. 7.After removing all unused User Accounts, restart the computer. Windows Updates Make sure we have all newest Windows Updates installed by visiting update.microsoft.com and following the instructions found on the website.
11. After 30 seconds 5 minutes you will be loaded into windows safe mode (it claims safe mode on all 4 corners of the screen). Windows may ask should you if you would like to enter secure mode. Choose Yes.
For more information about AVG anti-virus software, feel free to pay a visit to the AVG free anti-virus download home page. Make sure your computer is compatible before ever creating changes with it. Many protection programs and software are not compatible with each alternative, thus be sure and just create changes to your computer when you know what you may be doing. If you do not understand what we are doing I would advise having a computer technician create the changes for you.