Some theorems in accepted topology
Every bankrupt breach in R of bound breadth is compact. More is true: In Rn, a set is bunched if and alone if it is bankrupt and bounded. (See Heine–Borel theorem).
Every affiliated angel of a bunched amplitude is compact.
Tychonoff's theorem: the (arbitrary) artefact of bunched spaces is compact.
A bunched subspace of a Hausdorff amplitude is closed.
Every affiliated bijection from a bunched amplitude to a Hausdorff amplitude is necessarily a homeomorphism.
Every arrangement of credibility in a bunched metric amplitude has a allied subsequence.
Every breach in R is connected.
Every bunched finite-dimensional assorted can be anchored in some Euclidean amplitude Rn.
The affiliated angel of a affiliated amplitude is connected.
Every metric amplitude is paracompact and Hausdorff, and appropriately normal.
The metrization theorems accommodate all-important and acceptable altitude for a cartography to appear from a metric.
The Tietze addendum theorem: In a accustomed space, every affiliated real-valued action authentic on a bankrupt subspace can be continued to a affiliated map authentic on the accomplished space.
Any accessible subspace of a Baire amplitude is itself a Baire space.
The Baire class theorem: If X is a complete metric amplitude or a locally bunched Hausdorff space, again the autogenous of every abutment of countably abounding boilerplate close sets is empty.
On a paracompact Hausdorff amplitude every accessible awning admits a allotment of accord accessory to the cover.
Every path-connected, locally path-connected and semi-locally artlessly affiliated amplitude has a accepted cover.
General cartography additionally has some hasty access to added areas of mathematics. For example:
In cardinal theory, Fürstenberg's affidavit of the aeon of primes.
See additionally some counter-intuitive theorems, e.g. the Banach–Tarski one.
Every bankrupt breach in R of bound breadth is compact. More is true: In Rn, a set is bunched if and alone if it is bankrupt and bounded. (See Heine–Borel theorem).
Every affiliated angel of a bunched amplitude is compact.
Tychonoff's theorem: the (arbitrary) artefact of bunched spaces is compact.
A bunched subspace of a Hausdorff amplitude is closed.
Every affiliated bijection from a bunched amplitude to a Hausdorff amplitude is necessarily a homeomorphism.
Every arrangement of credibility in a bunched metric amplitude has a allied subsequence.
Every breach in R is connected.
Every bunched finite-dimensional assorted can be anchored in some Euclidean amplitude Rn.
The affiliated angel of a affiliated amplitude is connected.
Every metric amplitude is paracompact and Hausdorff, and appropriately normal.
The metrization theorems accommodate all-important and acceptable altitude for a cartography to appear from a metric.
The Tietze addendum theorem: In a accustomed space, every affiliated real-valued action authentic on a bankrupt subspace can be continued to a affiliated map authentic on the accomplished space.
Any accessible subspace of a Baire amplitude is itself a Baire space.
The Baire class theorem: If X is a complete metric amplitude or a locally bunched Hausdorff space, again the autogenous of every abutment of countably abounding boilerplate close sets is empty.
On a paracompact Hausdorff amplitude every accessible awning admits a allotment of accord accessory to the cover.
Every path-connected, locally path-connected and semi-locally artlessly affiliated amplitude has a accepted cover.
General cartography additionally has some hasty access to added areas of mathematics. For example:
In cardinal theory, Fürstenberg's affidavit of the aeon of primes.
See additionally some counter-intuitive theorems, e.g. the Banach–Tarski one.
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