Group: philosophy of mathematics
Group: sets
Topic: event time
Topic: geometry
Topic: Petri net
 
Summary
A topological property is one that is invariant under homeomorphisms, i.e. continuous, bijective maps with bijective inverses. Topology is the study of homeomorphic spaces. Informally, topology is the study of spaces in which an object can be arbitrarily deformed as long as it is not pinched or torn apart. A simple topology is the power set of all subsets of a finite set. (cbb 4/94)
Subtopic: general relativity as topology
Quote: continuous transformations of 4d coordinates express the topological order of points; i.e., neighboring points have nearly the same coordinates
 Subtopic: state space as topology
Quote: a state space of regions and boundaries forms a topology of open and closed points; isomorphic with undirected Petri nets [»holtAW11_1980]
 Subtopic: subset topology
Quote: a class of subsets is a topology if it is closed under arbitrary unions and finite intersections [»simmGF_1963]
 Quote: the discrete topology of a set is the class of all of its subsets; it is the strongest possible topology [»simmGF_1963]
 Subtopic: open set topology
Quote: there are many different ways of defining a topological space; the open set approach appears to be the simplest [»simmGF_1963]
 Quote: a subset of a metric space is open if every member belongs to an open sphere in the set [»simmGF_1963]
 Quote: open sets are closed under unions and finite intersections [»simmGF_1963]
 Quote: a neighborhood of a point is an open set that contains the point [»simmGF_1963]
 Quote: since metrics not needed for defining continuity, suggests defining open sets independently of metrics [»simmGF_1963]
 Subtopic: cell complex
Quote: describe a cartographic map by a cell complex of lines (1cells) with bounding points (0cells) and areas (2cells) [»honeSK_1986]
 Subtopic: representing topology
Quote: Sketchpad supports most straight edge and compass constructions; topological relationships via pseudo pen location and demonstrative language [»suthIE5_1963]
 Subtopic: topological maps
Quote: define the inverse of a function f by its properties on subsets of f's range [»simmGF_1963, OK]
 Quote: a map between topological spaces is continuous (open) if its inverse (it) maps open sets to open sets [»simmGF_1963]
 Quote: a mapping is continuous iff its inverse maps open sets to open sets [»simmGF_1963]
 Subtopic: homeomorphism
Quote: a topological property is a property that is preserved under homeomorphisms [»simmGF_1963]
 Quote: a homeomorphism is a onetoone, continuous open map between topological spaces [»simmGF_1963]
 Subtopic: dense and closed subsets
Quote: a subset of a metric space is closed if it contains all of its limit points; i.e., separated from its complement [»simmGF_1963]
 Quote: x is a limit point of a subset A of a metric space if every open sphere about x includes another element of A [»simmGF_1963]
 Quote: a closed set of a topological space is a set whose complement is open; preserved under intersection and finite unions [»simmGF_1963]
 Quote: the closure of a set is the intersection of all closed supersets; only closed sets preserved under closure [»simmGF_1963]
 Quote: a subset is dense in a topological space X if its closure is X [»simmGF_1963]
 Subtopic: separation
Quote: a topological space is separable if it contains a countable, dense subset [»simmGF_1963]
 Quote: separation properties of a topology are important because the supply of open sets is tied to the supply of continuous functions [»simmGF_1963]
 Quote: a topological space is T1 if every pair of distinct points separated by a neighborhood; implies points are closed [»simmGF_1963]
 Quote: a topological space is Hausdorff if every pair of distinct points have disjoint neighborhoods [»simmGF_1963]
 Quote: every compact subspace of a Hausdorff space is closed [»simmGF_1963]
 Quote: if continuous functions separate points, the space is Hausdorff [»simmGF_1963]
 Quote: the greater the INTERESTS of an ORGANIZATION the sharper the boundaries [»holtAW_1997]
 Quote: navigators distinguish open water and a harbor but not being in between the two; it does not matter

Related Topics
Group: philosophy of mathematics (11 topics, 330 quotes)
Group: sets (7 topics, 148 quotes)
Topic: event time (45 items)
Topic: geometry (33 items)
Topic: Petri net (44 items)
