mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, particularly

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, the K-topology is a

that one can impose on the set of all real numbers which has some interesting properties. Relative to the set of all real numbers carrying the

standard topology In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...

, the set ''K'' = is not

closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...

since it doesn't contain its (only) limit point 0. Relative to the K-topology however, the set K is automatically decreed to be closed by adding ‘more’ basis elements to the standard topology on R. Basically, the K-topology on R is strictly finer than the standard topology on R. It is mostly useful for counterexamples in basic topology.

Formal definition

Let R be the set of all real numbers and let ''K'' = . Generate a topology on R by taking

basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...

as all open intervals (''a'', ''b'') and all sets of the form (''a'', ''b'') – ''K'' (the set of all elements in (''a'', ''b'') that are not in ''K''). The

generated is known as the K-topology on R. The sets described in the definition form a basis (they satisfy the conditions to be a basis).

Properties and examples

Throughout this section, ''T'' will denote the K-topology and (R, ''T'') will denote the set of all real numbers with the K-topology as a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

. 1. The topology ''T'' on R is strictly finer than the standard topology on R but not comparable with the

lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of in ...

on R 2. From the previous example, it follows that (R, ''T'') is not

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

3. (R, ''T'') is Hausdorff but not regular. The fact that it is Hausdorff follows from the first property. It is not regular since the closed set K and the point have no disjoint

neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...

s about them 4. Surprisingly enough, (R, ''T'') is a connected topological space. However, (R, ''T'') is not

path connected In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...

; it has precisely two path components: (−∞, 0] and (0, +∞) 5. (R, ''T'') is not Locally connected space, locally path connected (since its path components are not equal to its

components Circuit Component may refer to: •Are devices that perform functions when they are connected in a circuit. In engineering, science, and technology Generic systems * System components, an entity with discrete structure, such as an assem ...

). It is also not

locally connected In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness a ...

at but it is locally connected everywhere else 6. The closed interval ,1is not compact as a subspace of (R, ''T'') since it is not even

limit point compact In mathematics, a topological space ''X'' is said to be limit point compact or weakly countably compact if every infinite subset of ''X'' has a limit point in ''X''. This property generalizes a property of compact spaces. In a metric space, limit ...

(''K'' is an infinite subspace of ,1that has no limit point in ,1 7. In fact, no subspace of (R, ''T'') containing ''K'' can be compact. If ''A'' were a subspace of (R, ''T'') containing ''K'', ''K'' would have no limit point in ''A'' so that ''A'' can not be limit point compact. Therefore, ''A'' cannot be compact 8. The quotient space of (R, ''T'') obtained by collapsing ''K'' to a point is not Hausdorff. ''K'' is distinct from 0, but can't be separated from 0 by disjoint open sets.

References

* {{cite book , author = James Munkres , author-link = James Munkres , year = 1999 , title = Topology , edition = 2nd , publisher =

Prentice Hall Prentice Hall was an American major educational publisher owned by Savvas Learning Company. Prentice Hall publishes print and digital content for the 6–12 and higher-education market, and distributes its technical titles through the Safari B ...

, isbn = 0-13-181629-2 Topological spaces

Formal definition

Properties and examples

See also

References