Self-based space - Revision history

Vipul: 2 revisions

2008-05-11T19:57:59Z

2 revisions

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Vipul: /* Metaproperties */

2007-12-17T19:52:24Z

Metaproperties

← Older revision		Revision as of 19:52, 17 December 2007
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	==Metaproperties==		==Metaproperties==

	{{~~finite~~ DP-closed}}		{{DP-closed}}

	~~A finite~~ product ~~of self-based spaces is self~~-~~based. In fact, we can take as the new basis, the rectangles obtained from basis elements in each of the factors.~~		{{box-product-closed}}

	{{open subspace-closed}}		{{open subspace-closed}}

	An [[open subset]] of a self-based space is self-based under the induced subspace topology. The idea is that we can choose a ''smaller'' basis element about each point in the subspace, that is homeomorphic to the open subset.		An [[open subset]] of a self-based space is self-based under the induced subspace topology. The idea is that we can choose a ''smaller'' basis element about each point in the subspace, that is homeomorphic to the open subset.

Vipul at 21:27, 27 October 2007

2007-10-27T21:27:56Z

New page

{{topospace property}}

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==Definition==

A [[topological space]] is said to be '''self-based''' if it has a basis of open sets for which every basis set is homeomorphic to the whole space.

==Relation with other properties==

===Stronger properties===

* [[Euclidean space]]

===Weaker properties===

* [[Uniformly based space]]

==Metaproperties==

{{finite DP-closed}}

A finite product of self-based spaces is self-based. In fact, we can take as the new basis, the rectangles obtained from basis elements in each of the factors.

{{open subspace-closed}}

An [[open subset]] of a self-based space is self-based under the induced subspace topology. The idea is that we can choose a ''smaller'' basis element about each point in the subspace, that is homeomorphic to the open subset.