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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 ${\displaystyle K=\{1/n:n=1,2,\dots \}}$ 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 ${\displaystyle K=\{1/n:n=1,2,\dots \}.}$ The K-topology on R is the topology obtained by taking as a base the collection of all open intervals ${\displaystyle (a,b)}$ together with all sets of the form ${\displaystyle (a,b)\setminus K.}$ The neighborhoods of a point ${\displaystyle x\neq 0}$ are the same as in the usual Euclidean topology. The neighborhoods of ${\displaystyle 0}$ are of the form ${\displaystyle V\setminus K}$, where ${\displaystyle V}$ is a neighborhood of ${\displaystyle 0}$ in the usual topology. The open sets in the K-topology are precisely the sets of the form ${\displaystyle U\setminus B}$ with ${\displaystyle U}$ open in the usual Euclidean topology and ${\displaystyle B\subseteq K.}$

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 ${\displaystyle 0}$, but the set ${\displaystyle K}$ and the point ${\displaystyle 0}$ 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: ${\displaystyle (-\infty ,0]}$ and ${\displaystyle (0,+\infty ).}$

4. The K-topology is not locally path connected at ${\displaystyle 0}$ and not locally connected at ${\displaystyle 0}$. But it is locally path connected and locally connected everywhere else.

5. The closed interval 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 ). More generally, no subspace A of (R, T) containing K is compact.

See also

• List of topologies

Notes

1. ^ Munkres 2000, p. 82.
2. ^ Steen & Seebach 1995, Counterexample 64.
3. Willard 2004, Example 14.2.

References

• Munkres, James R. (2000). Topology (2nd ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260.
• Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) . Counterexamples in Topology (Dover reprint of 1978 ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-486-68735-3. MR 0507446.
• Willard, Stephen (2004) . General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.

• Topological spaces