K-topology
Encyclopedia
In mathematics
, particularly topology
, the K-topology is a topology
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, the set K = {1/n | n is a natural number
} is not closed
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.
generated is known as the K-topology on R.
Note that: The sets described in the definition do form a basis (they satisfy the conditions to be a basis).
.
1. The topology T on R is strictly finer than the standard topology on R but not comparable with the lower limit topology
on R
2. From the previous example, it follows that (R, T) is not compact
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 {0} have no disjoint neighbourhoods
about them
4. Surprisingly enough, (R, T) is a connected topological space
. However, (R, T) is not path connected
; it has precisely two path components
: (−∞, 0] and (0, +∞)
5. Note also that (R, T) is not locally path connected
(since its path components are not equal to its components
). It is also not locally connected
at {0} but it is locally connected everywhere else
6. 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 subspace of [0,1] that has no limit point in [0,1])
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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, the K-topology is a topology
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
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, the set K = {1/n | n is a natural number
Natural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
} is not closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
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
Basis
Basis may refer to* Cost basis, in income tax law, the original cost of property adjusted for factors such as depreciation.* Basis of futures, the value differential between a future and the spot price...
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 = {1/n | n is a natural number}. Generate a topology on R by taking as basis 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 topologyTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
generated is known as the K-topology on R.
Note that: The sets described in the definition do 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 spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
.
1. The topology T on R is strictly finer than the standard topology on R but not comparable with the lower limit topology
Lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties...
on R
2. From the previous example, it follows that (R, T) is not compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
3. (R, T) is Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
but not regular
Regular space
In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...
. The fact that it is Hausdorff follows from the first property. It is not regular since the closed set K and the point {0} have no disjoint neighbourhoods
Neighbourhood
A neighbourhood or neighborhood is a geographically localised community within a larger city, town or suburb. Neighbourhoods are often social communities with considerable face-to-face interaction among members. "Researchers have not agreed on an exact definition...
about them
4. Surprisingly enough, (R, T) is a connected topological space
Connected space
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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
. However, (R, T) is not path connected
Connected space
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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
; it has precisely two path components
Locally connected space
In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...
: (−∞, 0] and (0, +∞)
5. Note also that (R, T) is not locally path connected
Locally connected space
In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...
(since its path components are not equal to its components
Locally connected space
In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...
). It is also not locally connected
Locally connected space
In topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...
at {0} but it is locally connected everywhere else
6. The closed interval [0,1] is not compact as a subspace of (R, T) since it is not even limit point compact
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 point compactness, compactness, and sequential compactness are...
(K is an infinite subspace of [0,1] that has no limit point in [0,1])
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
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of (R, T) obtained by collapsing K to a point is not Hausdorff
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.
See also
- Lower limit topologyLower limit topologyIn mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties...
- Natural topologyNatural topologyIn any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question...
- SequenceSequenceIn mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
- Locally connected spaceLocally connected spaceIn topology and other branches of mathematics, a topological space X islocally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.-Background:...
- Connected spaceConnected spaceIn 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 nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...