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Topic: Subbasis

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	Talk:Subbase - Wikipedia, the free encyclopedia
		the intersection of the empty subcollection of the subbasis, you get X. A good explanation of this is at nullary intersection.
		Fine, you say the subbasis "generates" X, in the sense that the generated topology is the smallest containing the subbasis.
		But, if the subbasis does not cover X, then the "constructive" definition (unions of finite intersection, without the convention of nullary intersection) does not give the topology generated by X.
	en.wikipedia.org /wiki/Talk:Subbase (1041 words)

	PlanetMath: subbasis
		This is version 5 of subbasis, born on 2002-01-06, modified 2004-06-22.
		Object id is 1409, canonical name is Subbasis.
		I think that condition that the union of A is the whole X should be required in the second paragraph.
	www.planetmath.org /encyclopedia/Subbasis.html (75 words)

	Subbase - Wikipedia, the free encyclopedia
		In highway engineering, subbase is a layer between subgrade and the base course.
		In topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T which generates T, in the sense that T is the smallest topology containing B.
		In general, however, there is no unique subbasis for a given topology.
	en.wikipedia.org /wiki/Subbase (1057 words)

	Tychonoff's theorem - Wikipedia, the free encyclopedia
		Finally, it's possible to show that if one of the spaces in the product has a subbasis element containing that space's coordinate of x, then the "tube" formed by pulling that subbasis into the full product space with an inverse projection map will contain x and will also intersect every element of D.
		But tubes form a subbasis in the product topology, and so, also by Lemma 2, all basis elements containing x are in D.
		But then these basis elements intersect every element of D, and so x is a limit point of each element of D, and so is in the closure of each element of D.
	en.wikipedia.org /wiki/Tychonoff%27s_theorem (1201 words)

	sir~~topology problem
		It isn't a basis because there is a point in X which is not in an element of A. But that doesn't mean that A can't _generate_ a topology.
		Are you sure you understand the difference between a basis and a subbasis?
		Every collection of subsets of a set X is a subbasis for some topology on X.
	www.forum-one.org /new-5211383-4346.html (303 words)

	tychonoff's theorem (Site not responding. Last check: 2007-10-20)
		But we know these spaces are compact, and so we can choose a point in each space from the intersection of that space's projected
		Finally, it's possible to show that if one of the spaces in the product has a subbasis element containing that space's coordinate of
		But tubes form a subbasis in the product topology, and so, also by Lemma 2, all basis elements containing
	www.yourencyclopedia.net /Tychonoff%27s_theorem.html (805 words)

	t u t o r i a l @ r e s
		A component set is relative to the basis and subbasis chosen for the tour code.
		Where no basis and subbasis exist a standard set will be created as soon as the tour code is typed into a new booking.
		Where they DO exist, the appropriate component set will be created when the basis and subbasis are chosen.
	www.respax.com.au /R4W/Tutorial/Tut_PkgSetup.shtml (290 words)

	Untitled Document
		Where pax and basis has been selected, the basis field must be configured and filled in but the subbasis field will remain grey (not valid), and where pax, basis and subbasis has been selected, all fields must be configured and filled in.
		Where basis and/or subbasis are being used in the price matrix - they must be created and configured using the edit basis button.
		Tick this box where a levy is applicable to the tour, basis and subbasis.
	www.respax.com.au /R4W/help/Maint_PriceMatrix.htm (474 words)

	Algebraic Topology: Topology
		A basis for the open sets is a collection of open sets such that each open set is a union of some subcollection.
		A subbasis for the open sets is a collection of open sets such that one obtains a basis by taking finite intersections.
		In order to check that a given function is continuous, it suffices to check that the inverse images of the members of a subbasis for the open sets are open again.
	www.win.tue.nl /~aeb/at/algtop-2.html (1509 words)

	LatexWiki Website - EasyProofOfTychonoff (Site not responding. Last check: 2007-10-20)
		Lemma (Alexander): If X is a topological space and S is a subbasis of X, then X is compact if and only if every S-cover of X has a finite subcover.
		Since G was an arbitrary S-cover, the conditions of Alexander's subbase lemma are satisfied, and X is compact.
		Proof of Alexander subbasis lemma: Let B be the collection of all open covers of X that do not have finite subcovers, B for bad covers.
	latexwiki.rootnode.com /wiki/EasyProofOfTychonoff (434 words)

	iqexpand.com (Site not responding. Last check: 2007-10-20)
		pl:podbaza In topology, a subbase (or subbasis) for a topological space X with topology T is a subcollection B of T such that the collection of finite intersections of elements of B form a basis for T.
		There is, in general, no unique subbasis for a given topology.
		For any subcollection S of the power set P(X), there is a unique smallest topology T containing S (namely, the arbitrary unions of the finite intersections of members of S).
	subbase.iqexpand.com (794 words)

	Orderability of Topological Spaces, by Steve Purisch (Site not responding. Last check: 2007-10-20)
		Rudin's problems on monotone normality, Topology Atlas Invited Contributions, vol.
		van Dalen and E. Wattel characterized orderable spaces using a subbasis property in: A topological characterization of ordered spaces, Gen. Top.
		However, the subbase induces a unique order up to inverse on the space.
	at.yorku.ca /z/a/a/b/14.htm (406 words)

	subbasis - OneLook Dictionary Search (Site not responding. Last check: 2007-10-20)
		We found 3 dictionaries with English definitions that include the word subbasis:
		Tip: Click on the first link on a line below to go directly to a page where "subbasis" is defined.
		Subbasis : Eric Weisstein's World of Mathematics [home, info]
	www.onelook.com /?w=subbasis (77 words)

	[No title]
		A set S of open sets is called a subbasis for the topology if every open set is a union of finite intersections of sets in S. Of course the word ``finite'' here does not mean that the intersection should be a finite set, but that is the intersection of finitely many sets.
		With these conventions we then have that if X is a set and S and arbitrary set of parts of X, there is exactly one topology T(S) on X such that S is a subbasis for T(S) (the topology generated by S).
		S is a subbasis for the topology of X if and only if n B = { / s_i such that s_i,..., s_n is in S} i=1 is a basis for the topology of X. is this correct?
	www.grahamkendall.net /Math/MathNewsgroups/mm-499.txt (2076 words)

	Untitled Document
		Tour basis is designed to extend the flexibility of the price matrix as well as provide an easy way to change pickup lists and options for a tour without having to set up a new tour code.
		You must have configured the price matrix type in tour maintenance to be either pax, pax and basis or pax, basis and subbasis to be able to utilise the tour basis and subbasis.
		Makes this subbasis the automatic selection when tabbing into a blank type field on the reservations new booking form.
	www.respax.com.au /ROO/data/help/reference/Maint_Basis.htm (842 words)

	t u t o r i a l @ r e s
		The way that the Basis and Sub Basis are designed to be used for Accommodation is more standardised than for tours and packages.
		It is here that you must set the number of passengers per subbasis (see the image above).
		As with tours and packages, you can select which basis will be the default, and which sub basis will be the default, so the most commonly used rooms are automatically selected first.
	www.respax.com /R4W/Tutorial/Tut_AccomSetup.shtml (177 words)

	t u t o r i a l @ r e s
		When making a booking, the res staff can choose if the booking is allotment booking by checking the allotment booking checkbox in the reservation screen.
		Prices can be configured specifically for a date range, days, basis, subbasis and pax types.
		Before you can setup the price matrix, you must select a price matrix type and create basis and subbasis items for this tour code.
	www.respax.com.au /R4W/Tutorial/Tut_TourSetup.shtml (696 words)

	topology.html
		A subbasis B is used to generate a topology by closing B under both finite intersections and finite unions and adding X and {}.
		Since X is a finite set the result is a topology.
		Procedure to find a subbasis for a product topology using inverse projections.
	www.adeptscience.co.uk /maplearticles/f879.html (544 words)

	Subbases of a Topology
		Given that I know a topology T of a set X and a subbasis S of T. Is it
		and a natural subbasis which is much smaller.
		Now had S been a base, this would be true wouldn't it?
	www.groupsrv.com /science/post-928321.html (558 words)

	Citations: AMS Colloquium Publications - Birkhoff (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		The interval topology on L is defined by taking the intervals [ p, #)and(#,q] for all p, q # L as a subbasis of closed sets.
		We shall establish a similar result for sublattices of topological lattice A(X) endowed with the pointwise topology.
		First note that sets (a, b, #) and (a, #,b] form a subbasis....
	citeseer.ist.psu.edu /context/446182/0 (2703 words)

	MathLinks Math Forum :: View topic - topology problem(homemorph)
		has a subbasis of closed sets formed by sets (the straight lines) with the property that each two intersect in at most one point (this means that each closed set can be written as a finite union of arbitrary intersections of lines).
		This must be a finite union of intersections of sets in ur subbasis for
		This means that our set, which is an intersection of elements in the subbasis, contains the set
	www.mathlinks.ro /Forum/post-215241.html (431 words)

	All finite distributive lattices occur as intervals between Hausdorff topologies
		denote the least cardinal of a filter subbasis for a free ultrafilter on
		is too small to be an ultrafilter subbasis, so there exist disjoint sets
		is too small to be an ultrafilter subbasis, there exist disjoint sets
	pear.math.pitt.edu /mathzilla/Examples/svg.html (830 words)

	Introductory Topology
		subbasis, product topology (2 factors), closure and interior
		Applications of fundamental group of the circle, Fundamental Theorem of Algebra
		Classification of covering spaces by subgroups of the fundamental group
	www.faculty.iu-bremen.de /stoll/teaching/IntroTopology-2005-Spring/index.html (29 words)

	Tychonoff's theorem (Site not responding. Last check: 2007-10-20)
		then the "tube" formed by pulling subbasis into the full product space with inverse projection map will contain
		But tubes form a subbasis in product topology and so also by Lemma all basis elements containing
		But then these basis elements intersect element of
	www.freeglossary.com /Tychonoff%27s_theorem (1201 words)

	725 Assignments (Site not responding. Last check: 2007-10-20)
		Prove that the topology generated by a subbasis is the intersection of all topologies containing the subbasis.
		Find a metric space X and a limit point x in X with a homeomorphism between X and X--{x}.
		Find an example of one that is not normal.
	www.math.wichita.edu /~pparker/classes/725assgn.htm (178 words)

	Topics: Topology, Topological Space
		On a Lorentzian manifold: Use the Alexandrov topology, or for compact cases Johan's definition.
		On a poset: Interval topology (closed intervals are a subbasis for closed sets); Order topology (G
		P is open if G eventually contains any net P which order-converges to an element of G, @ Birkhoff 67).
	www.phy.olemiss.edu /~luca/Topics/t/top.html (539 words)

	Topology Course Lecture Notes
		The second stage is describing what open sets look like.
		This will be done by describing a subbasis for the topology.
		The guiding goal is to provide just enough opens sets to guarantee the continuity of certain important functions.
	at.yorku.ca /i/a/a/b/23.dir/index.htm (8277 words)

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