Limit point compact

In mathematics, particularly topology, limit point compactness is a certain condition on a topological space which generalizes some features of compactness. In a metric space, limit point compactness, compactness, and sequential compactness are all equivalent. For general topological spaces, however, these three notions of compactness are mutually inequivalent.

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.

Properties and Examples

* Limit point compactness is equivalent to countable compactness if X is a T1-space and is equivalent to compactness if X is a metric space.

* An easy example of a space X that is not weakly countably compact is any countable (or larger) set with the discrete topology. A more interesting example is the countable complement topology.

* Even though a continuous function from a compact space "X", to an ordered set "Y" in the order topology, must be bounded, the same thing does not hold if "X" is "limit point compact". An example is given by the space X imesmathbb{Z} (where "X" = {1, 2} carries the indiscrete topology and mathbb{Z} is the set of all integers carrying the discrete topology) and the function f=pi_{mathbb{Z given by projection onto the second coordinate. Clearly, "f" is continuous and X imes mathbb{Z} is limit point compact (in fact, "every" nonempty subset of X imes mathbb{Z} has a limit point) but "f" is not bounded, and in fact f(X imes mathbb{Z})=mathbb{Z} is not even limit point compact.

* Every countably compact space is weakly countably compact, but the converse is not true.

* For metrizable spaces, compactness, limit point compactness, and sequential compactness are all equivalent.

* The set of all real numbers is not limit point compact; the integers are an infinite set but do not have a limit point in mathbb{R}.

*If ("X", "T") and ("X", "T*") are topological spaces with "T*" finer than "T" and ("X", "T*") is limit point compact, then so is ("X", "T")

*A finite space is vacuously limit point compact

ee also

*Compact space
*Sequential compactness
*Metric space
*Bolzano-Weierstrass theorem

*planetmath|id=1234|title=Limit point compact
*planetmath|id=6212|title=Weakly countably compact

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Compact space — Compactness redirects here. For the concept in first order logic, see compactness theorem. In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness… …   Wikipedia

  • Limit set — In mathematics, especially in the study of dynamical systems, a limit set is the state a dynamical system reaches after an infinite amount of time has passed, by either going forward or backwards in time. Limit sets are important because they can …   Wikipedia

  • Compact operator — In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y. Such an… …   Wikipedia

  • Compact star — In astronomy, the term compact star (sometimes compact object) is used to refer collectively to white dwarfs, neutron stars, other exotic dense stars, and black holes. These objects are all small for their mass. The term compact star is often… …   Wikipedia

  • Point system (driving) — A demerit point system is one in which a driver s licensing authority, police force, or other organization issues cumulative demerits, or points to drivers on conviction for road traffic offenses. Points may either be added or subtracted,… …   Wikipedia

  • Point process — In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces. For example, the occurrence …   Wikipedia

  • compact — I adj 1. dense, solid, firm, hard; close, tight, tight knit, close knit, impermeable, impassable, impenetrable; thick, heavy, thickset, stubby, squat, short, stocky; condensed, concentrated, compressed, packed together, squeezed together, crowded …   A Note on the Style of the synonym finder

  • Countably compact space — In mathematics a topological space is countably compact if every countable open cover has a finite subcover. Examples and Properties A compact space is countably compact. Indeed, directly from the definitions, a space is compact if and only if it …   Wikipedia

  • Sequentially compact space — In mathematics, a topological space is sequentially compact if every sequence in the space has a convergent subsequence. Examples and Properties1. A metrizable space is sequentially compact if and only if it is compact. However, in general there… …   Wikipedia

  • Particular point topology — In mathematics, the particular point topology (or included point topology) is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.