zero dimensional

Definition 1.

[1, 2]Suppose $X$ is a topological space. If $X$ has a basis consising ofclopen sets, then $X$ is said to be .

Examples of zero-dimensional spaces are: the set $\\mathbb{Q}$ of rational numbers (with subspace topology induced from the usual metric topology on $\\mathbb{R}$ , the set of real numbers), the Cantor space, as well as the Sorgenfrey line.

The concepts of zero-dimentionality and total disconnectedness are closely related. Indeed, every zero-dimentional $T_{1}$ space (http://planetmath.org/T1Space) is totally disconnected. Furthermore, if a topological space is locally compact and Hausdorff, then the notions of zero-dimentionality and total disconnectedness are equivalent.

References

1 L.A. Steen, J.A.Seebach, Jr.,Counterexamples in topology,Holt, Rinehart and Winston, Inc., 1970.
2 S. Willard, General Topology,Addison-Wesley, Publishing Company, 1970.

单词	ZeroDimensional
释义	zero dimensional Definition 1. [1, 2]Suppose $X$ is a topological space. If $X$ has a basis consising ofclopen sets, then $X$ is said to be . Examples of zero-dimensional spaces are: the set $\\mathbb{Q}$ of rational numbers (with subspace topology induced from the usual metric topology on $\\mathbb{R}$ , the set of real numbers), the Cantor space, as well as the Sorgenfrey line. The concepts of zero-dimentionality and total disconnectedness are closely related. Indeed, every zero-dimentional $T_{1}$ space (http://planetmath.org/T1Space) is totally disconnected. Furthermore, if a topological space is locally compact and Hausdorff, then the notions of zero-dimentionality and total disconnectedness are equivalent. References 1 L.A. Steen, J.A.Seebach, Jr.,Counterexamples in topology,Holt, Rinehart and Winston, Inc., 1970. 2 S. Willard, General Topology,Addison-Wesley, Publishing Company, 1970.
随便看	Laplace transform of {f⁢(t)}/t LaplaceTransformOfIntegral Laplace transform of integral LaplaceTransformOfLogarithm Laplace transform of logarithm LaplaceTransformOfPeriodicFunctions Laplace transform of periodic functions LaplaceTransformOfPowerFunction Laplace transform of power function LaplaceTransformOfSineIntegral Laplace transform of sine integral LaplaceTransformOfTnft Laplace transform of t^n⁢f⁢(t) LaplaceTransformsOfDerivatives Laplace transforms of derivatives Laplacian Laplacian Laplacian from Rectangular to Spherical Coordinates LaplacianFromRectangularToSphericalCoordinates1 LaplacianMatrixOfAGraph Laplacian matrix of a graph LargeIdeal large ideal LargeIntegersThatAreOrMightBeTheSmallestOfTheirKind large integers that are or might be the smallest of their kind