فضاء صفري البعد

هذا المقال هو عن zero dimension in topology. إذا كنت تريد several kinds of zero space in algebra، انظر zero object (algebra).

في الرياضيات، الفضاء الطبولوجي صفري البعد ( zero-dimensional topological space أو nildimensional space) هو فضاء طبولوجي له البعد صفر فيما يتعلق بواحد من several inequivalent notions of assigning a dimension to a given topological space. A graphical illustration of a nildimensional space is a point.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 التعريف

تحديداً:

• A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement which is a cover by disjoint open sets.
• A topological space is zero-dimensional with respect to the finite-to-finite covering dimension if every finite open cover of the space has a refinement that is a finite open cover such that any point in the space is contained in exactly one open set of this refinement.
• A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets.

The three notions above agree for separable, metrisable spaces.

 خصائص الفضاءات ذات بعد صفري منخفضة الحث

• A zero-dimensional Hausdorff space is necessarily totally disconnected, but the converse fails. However, a locally compact Hausdorff space is zero-dimensional if and only if it is totally disconnected. (See (Arhangel'skii & Tkachenko 2008, Proposition 3.1.7, p.136) for the non-trivial direction.)
• Zero-dimensional Polish spaces are a particularly convenient setting for descriptive set theory. Examples of such spaces include the Cantor space and Baire space.
• Hausdorff zero-dimensional spaces are precisely the subspaces of topological powers $2^{I}$ where $2=\{0,1\}$ is given the discrete topology. Such a space is sometimes called a Cantor cube. If I is countably infinite, $2^{I}$ is the Cantor space.

 الطيات

All points of a zero-dimensional manifold are isolated.

 الكرة الفائقة

The zero-dimensional hypersphere (0-sphere) is a pair of points, and the zero-dimensional ball is a single point.

 ملاحظات

• Arhangel'skii, Alexander; Tkachenko, Mikhail (2008). Topological Groups and Related Structures. Atlantis Studies in Mathematics. Vol. 1. Atlantis Press. ISBN 978-90-78677-06-2.
• Engelking, Ryszard (1977). General Topology. PWN, Warsaw.
• Willard, Stephen (2004). General Topology. Dover Publications. ISBN 0-486-43479-6.