Ads
related to: geometry counterexamples
Search results
Results from the WOW.Com Content Network
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem , topologists (including Steen and Seebach) have defined a wide variety of topological properties .
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property .
In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. [1] For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are ...
Therefore, it serves as an important counterexample in topology. [1] Intuitively, the usual real-number line consists of a countable number of line segments [ 0 , 1 ) {\displaystyle [0,1)} laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.
One famous counterexample in topology is the Alexander horned sphere, showing that topologically embedding the sphere S 2 in R 3 may fail to separate the space cleanly. As a counterexample, it motivated mathematicians to define the tameness property, which suppresses the kind of wild behavior exhibited by the horned sphere, wild knot , and ...
Counterexamples: Borsuk found a compact subset of that is an ANR but not strictly locally contractible. [13] (A space is strictly locally contractible if every open neighborhood of each point contains a contractible open neighborhood of .) Borsuk also found a compact subset of the Hilbert cube that is locally contractible (as defined above) but ...
Ads
related to: geometry counterexamples