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Topic: Subspace (topology)

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Subspace (topology)

Connected space

Homeomorphism

	PlanetMath: subspace topology
		is the topology whose open sets are those subsets of
		obtained by taking the subspace topology is called a topological subspace, or simply subspace, of
		This is version 3 of subspace topology, born on 2001-10-25, modified 2003-03-13.
	planetmath.org /encyclopedia/SubspaceTopology.html (45 words)

	Topology Encyclopedia
		Topology (Greek topos, "place," and logos, "study") is a branch of mathematics that is an extension of geometry.
		The most basic division within topology is into point-set topology, which investigates such concepts as compactness, connectedness, countability, and algebraic topology, which investigates such concepts as homotopy, homology, and knot theory.
		In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
	www.hallencyclopedia.com /topic/Topology.html (1819 words)

	PlanetMath: topological space
		Cross-references: metric topology, product topology, subspace topology, indiscrete topology, power set, discrete topology, coarser, finer, complement, closed set, subsets
		Can a topology be defined as a subset of an arbitrary complete (and complemented) lattice, instead of a power set?
		it might be interesting to note that you can equivalently define a topology in terms of it's closed sets, by demorgan's set laws.
	planetmath.org /encyclopedia/TopologicalSpace.html (302 words)

	The subspace topology
		R (with its usual topology/metric) is the discrete topology.
		The subspace topology on the x-axis as a subset of R
		X then the subspace topology is the weakest topology (fewest open sets) on A in which this map is continuous.
	www-history.mcs.st-and.ac.uk /~john/MT4522/Lectures/L14.html (154 words)

	PlanetMath: order topology
		The subspace topology is always finer than the induced order topology, but they are not in general the same.
		Cross-references: separating, disjoint, without loss of generality, points, Hausdorff, chain, contain, open, singleton, finer, subspace, subspace topology, order, induced, subset, standard topologies, open intervals, basis, equivalent, open rays, subbasis, generated by, topology, linearly ordered set
		This is version 6 of order topology, born on 2002-01-06, modified 2007-01-19.
	planetmath.org /encyclopedia/OrderTopology.html (193 words)

	Topology Course Lecture Notes
		We learnt that, for metric spaces, sequential convergence was adequate to describe the topology of such spaces (in the sense that the basic primitives of `open set', `neighbourhood', `closure' etc. could be fully characterised in terms of sequential convergence).
		This topology is 'just right' in the sense that it is barely fine enough to guarantee the continuity of the coordinate projection functions while being just course enough allow the important result of Theorem.
		A basic formal distinction between algebra and topology is that although the inverse of a one-one, onto group homomorphism [etc!] is automatically a homomorphism again, the inverse of a one-one, onto continuous map can fail to be continuous.
	at.yorku.ca /i/a/a/b/23.dir (8277 words)

	Mathematics Algebraic Topology Homework Help
		A certain topology of X contains (among others) the sets {a}, {b}, and {c}.
		Subspace Topology: Interior, Closure, Boundary and Limit Points
		The subspaces X and Y of (SYMBOL) inherit the subspace topology.
	www.brainmass.com /homework-help/math/algebraic-topology (304 words)

	The subspace topology
		R (with its usual topology/metric) is the discrete topology.
		The subspace topology on the x-axis as a subset of R
		X then the subspace topology is the weakest topology (fewest open sets) on A in which this map is continuous.
	www-groups.dcs.st-and.ac.uk /~john/MT4522/Lectures/L14.html (154 words)

	General Topology - NoiseFactory Science Archives (http://noisefactory.co.uk)
		Unless stated otherwise, subsets are always assumed to carry this subspace topology, and are then called subspaces.
		If we assign P the discrete topology, in which every subset is open, these will include all the inverse images of open sets in the various factor spaces.
		The standard topologies on N, Z, Q, and R are all (defined to be) their order topologies.
	noisefactory.co.uk /maths/topology.html (4788 words)

	[No title]
		Topology Glossary Mainly extracted from (a) UC Davis Math:Profile Glossary (http://www.math.ucdavis.edu/profiles/glossary.html) by Greg Kuperberg (http://www.math.ucdavis.edu/profiles/kuperberg.html), and (b) Topology Atlas Glossary (http://www.achilles.net/~mtalbot/TopoGloss.html).
		An early result in topology states that every closed 3-manifold (closed meaning that the manifold is finite and connected but has no boundary) has a Heegaard splitting and a resulting description in terms of a Heegaard diagram, which describes how the two handlebodies are glued together.
		subspace A subset A of a topological space X with the inherited topology: the open set in A are the intersections of the open sets of X with A. support A straight path § in R2 supports a piecewise linear path w at s in (0,1) if §(1)=w(s) and w turns towards §(0) at s.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	More on Topology
		Topology is concerned with the study of the so-called topological properties of figures, that is to say properties that do not change under bicontinuous one-to-one transformations (called homeomorphisms).
		The root of topology was in the study of geometry in ancient cultures.
		"Topology", its English form, was introduced in print by Solomon Lefschetz in 1930 to replace the earlier name "analysis situs." The separate status of the topologist, a specialist in topology, was probably established from around 1920.
	www.artilifes.com /topology.htm (1524 words)

	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: )
		The cofinite topology (sometimes called the finite complement topology) is a topology which can be defined on every set X.
		This topology occurs naturally in the context of the Zariski topology.
		Subspaces: Every subspace topology of the cofinite topology is also the cofinite topology.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=cofinite (549 words)

	Quotient space Information
		In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space.
		We define a topology on the quotient set X/~ (the set consisting of all equivalence classes of ~) as follows: a set of equivalence classes in X/~ is open if and only if their union is open in X.
		Then the quotient topology on X/~ is the finest topology for which q is continuous.
	www.bookrags.com /wiki/Quotient_space (929 words)

	The Dispatch - Serving the Lexington, NC - News (Site not responding. Last check: )
		In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology).
		Given the real numbers with the usual topology the subspace topology of the natural numbers, as a subspace of the real numbers, is the discrete topology.
		The topology induced on a subset of a metric space by restricting the metric to this subset coincides with subspace topology for this subset.
	www.the-dispatch.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=subspace_topology (470 words)

	Topology MAT 530
		This is the largest (finest, strongest) topology such that the canonical projection (from the space to the quotient-space) is continuous.
		A counterexample is the set of all rational numbers with the topology induced from the reals (which is the same as the order topology) --- all rationals are separate connected components, but they are not open.
		Subspaces of Hausdorff spaces are Hausdorff, subspaces of regular spaces are regular.
	www.math.sunysb.edu /~timorin/mat530.html (2896 words)

	The Joel on Software Discussion Group - Geometry versus topology?
		In mathematics, a topology is something very formal, precise and quite abstract, based on a generalisation of the notion of a continous mapping.
		Perhaps "Topology" has a more general focus -- it's a 'ring' or a 'star' topology -- where the "Geometry" of a network is more specific to where the nodes are actually connected.
		It can be seen as the subspace topology taken when you embed the vertices (as points) and edges (as lines) of the graph in a euclidean space (often the plane R^2 when we're talking about planar graphs).
	discuss.joelonsoftware.com /default.asp?joel.3.441194.7 (1200 words)

	III. FCCR LIMITS
		A sufficient condition for a bounded operator not of finite rank to be of trace class, is that the spectral values have only one accumulation point, that it be zero, and that convergence to zero is sufficiently rapid that the sum analogous to (3.10) be bounded.
		What they are on the complementary subspace is undetermined except by the requirement of membership in Z. Therefore require that for every n: n-1 f> = SIGMA alpha_k k> (3.15) k=0 n-1 SIGMA (k + 1) alpha_k^2 < infinity (3.16) k=0 So, as n->infinity, ^2 must approach zero faster than n^(-2).
		The simple act of suppressing the subspace spanned by n, n-1>, is premature and not the only way of enacting the restriction as can be seen in the sections that deal with invariance and symmetry groups and with uncertainty relations.
	graham.main.nc.us /~bhammel/FCCR/III.html (2357 words)

	Topology
		The family t is called a topology (for X) when it satisfies these axioms and its elements are called _open sets_ (open wrt the topology).
		The reader should now check that continuity in the sense of calculus of a function from R to R is equivalent to continuity as a map of topological spaces, with respect to the topology m.
		Call a topology t _stronger_ than the topology t' (both for the same set X) if t is contained in t'.
	www.georgetown.edu /faculty/kainen/topology.html (1132 words)

	Subspaces in abstract Stone duality (Site not responding. Last check: )
		By abstract Stone duality we mean that the topology or contravariant powerset functor, seen as a self-adjoint exponential $\Sigma^{(-)}$ on some category, is monadic.
		Using Beck's theorem, this means that certain equalisers exist and carry the subspace topology.
		These subspaces are encoded by idempotents that play a role similar to that of nuclei in locale theory.
	www.tac.mta.ca /tac/volumes/10/13/10-13abs.html (248 words)

	XIII. UNCERTAINTY RELATIONS
		In [Zeeman 1967] it is argued that the physically correct topology on the lightcone should indeed be discrete.
		Consider the influence on equation (13.8b) by the restriction to a subspace, enforced in the n, k> basis.
		The problem is that the G(n)-positive subspace is invariant under neither the action of Q(n) nor the action of P(n).
	graham.main.nc.us /~bhammel/FCCR/XIII.html (1986 words)

	subspace download
		Although references to the existance of subspace may be found in both the comic and the cartoon.
		Continuum began as Sniper in 1995 and proceeded through 2 years of beta development as SubSpace and launched commercially on November continuum subspace download November 30.
		Subspace Note: the following is theory and conjecture..
	subspace-download.peekit.org (870 words)

	subspace
		Check the Continuum category for more subspace banner more Skins by this author and from other authors.
		If this is subspace continuum is your first visit.
		Although references to the existance of subspace may be found in both the comic and the cartoon..38 Full Version (5 MB) Download to play the latest public release of Continuum!
	subspace.peekit.net (915 words)

	Subspace topology@Everything2.com
		Suppose we have a topological space, X, its topology, T, and a subset of its points,
		Recall that the topology of a topological space is the collection of its open subsets, then the subspace topology on
		Hence all such sets are open in the subspace topology on
	www.everything2.com /index.pl?node_id=1089132 (178 words)

	PlanetMath: point
		point is a one-dimensional subspace of the vector space underlying the projective geometry.
		Anyone with an account can edit this entry.
		Cross-references: point-free geometry, primitive, geometry, topology, subspace, projective geometry, incidence geometry, affine space, vector space
	planetmath.org /encyclopedia/Point.html (226 words)

	ProvenMath (everything proven from axioms) - Apronus.com
		This page defines the concept of a topological subspace and provides all the basic theorems for dealing with subspaces.
		Interestingly, it is quite easy to convince oneself that the definition of the relative topology satisfies the axioms for a topological space.
		However, the easy straightforward proof uses the Axiom of Choice in a way that is almost transparent to an unexperienced eye.
	www.apronus.com /provenmath (985 words)

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