In mathematics, particularly in the field of topology, the K-topology, also called Smirnov's deleted sequence topology, is a topology on the set R of real numbers which has some interesting properties. Relative to the standard topology on R, the set K = { 1 / n : n = 1 , 2 , … } {\displaystyle K=\{1/n:n=1,2,\dots \}} is not closed since it doesn't contain its limit point 0. Relative to the K-topology however, the set K is declared to be closed by adding more open sets to the standard topology on R. Thus the K-topology on R is strictly finer than the standard topology on R. It is mostly useful for counterexamples in basic topology. In particular, it provides an example of a Hausdorff space that is not regular.
Formal definition Let R be the set of real numbers and let K = { 1 / n : n = 1 , 2 , … } . {\displaystyle K=\{1/n:n=1,2,\dots \}.} The K-topology on R is the topology obtained by taking as a base the collection of all open intervals ( a , b ) {\displaystyle (a,b)} together with all sets of the form ( a , b ) ∖ K . {\displaystyle (a,b)\setminus K.}
The neighborhoods of a point x ≠ 0 {\displaystyle x\neq 0} are the same as in the usual Euclidean topology. The neighborhoods of 0 {\displaystyle 0} are of the form V ∖ K {\displaystyle V\setminus K} , where V {\displaystyle V} is a neighborhood of 0 {\displaystyle 0} in the usual topology. The open sets in the K-topology are precisely the sets of the form U ∖ B {\displaystyle U\setminus B} with U {\displaystyle U} open in the usual Euclidean topology and B ⊆ K . {\displaystyle B\subseteq K.}
Properties 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. 1. The K-topology is strictly finer than the standard topology on R. Hence it is Hausdorff, but not compact. 2. The K-topology is not regular, because K is a closed set not containing 0 {\displaystyle 0} , but the set K {\displaystyle K} and the point 0 {\displaystyle 0} have no disjoint neighborhoods. And as a further consequence, the quotient space of the K-topology obtained by collapsing K to a point is not Hausdorff. This illustrates that a quotient of a Hausdorff space need not be Hausdorff. 3. The K-topology is connected. However, it is not path connected; it has precisely two path components: ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} and ( 0 , + ∞ ) . {\displaystyle (0,+\infty ).}
4. The K-topology is not locally path connected at 0 {\displaystyle 0} and not locally connected at 0 {\displaystyle 0} . But it is locally path connected and locally connected everywhere else. 5. The closed interval [0,1] is not compact as a subspace of (R, T) since it is not even limit point compact (K is an infinite closed discrete subspace of (R, T), hence has no limit point in [0,1]). More generally, no subspace A of (R, T) containing K is compact.
See also List of topologies
Notes
References Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. (accessible to patrons with print disabilities) Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978]. Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446. Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
