In mathematics, particularly in the field of topology, the K-topology,[1] also called Smirnov's deleted sequence topology,[2] is a topology on the set R of real numbers which has some interesting properties.
Relative to the standard topology on R, the set
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.
Let R be the set of real numbers and let
The K-topology on R is the topology obtained by taking as a base the collection of all open intervals
are the same as in the usual Euclidean topology.
open in the usual Euclidean topology and
[2] 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.
The K-topology is strictly finer than the standard topology on R. Hence it is Hausdorff, but not compact.
The K-topology is not regular, because K is a closed set not containing
And as a further consequence, the quotient space of the K-topology obtained by collapsing K to a point is not Hausdorff.
The K-topology is not locally path connected at
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.