Omega restricted space

In the mathematical sub-area of topology , the restriction denotes a weakening of the concept of compactness, which is central to the theory of topological spaces . ${\displaystyle \omega }$

definition

A topological space is called -bounded if every countable subset is contained in a compact subset. Often it is also required that the room is Hausdorff-like . We don't follow that here. ${\displaystyle \omega }$

Connection with other compactness terms

• Any compact space is limited. ${\displaystyle \omega }$
• Every restricted space is countably compact . ${\displaystyle \omega }$
• For metrizable spaces, the terms compact, limited and countable compact are all equivalent. In particular, -constraint is therefore synonymous with compactness for all subsets of the . ${\displaystyle \omega }$ ${\displaystyle \omega }$ ${\displaystyle \mathbb {R} ^{n}}$

Examples

• If one considers the ordinal number space with order topology , whereby the first uncountable ordinal number denotes, then this topological space is limited, but not compact. ${\displaystyle [0,\omega _{1})}$ ${\displaystyle \omega _{1}}$ ${\displaystyle \omega }$
• Likewise, the long straight line , which is closely related to the previous example, is limited but not compact. ${\displaystyle \omega }$

properties

• Closed subsets of -restricted spaces are -restricted. ${\displaystyle \omega }$ ${\displaystyle \omega }$
• Any products of -restricted spaces are -restricted. This is generally not true for countably compact spaces. ${\displaystyle \omega }$ ${\displaystyle \omega }$

Connection with 2-manifolds and the Bagpipe theorem

A 2-manifold is a Hausdorff space that is locally homeomorphic to the Euclidean plane . Usually, second countability is also required. Since such spaces can always be embedded in one , they are equivalent to -restriction and compactness. The -constraint (i.e. compact) 2-manifolds are all completely classified: Each of these is created from a sphere by sticking a finite number of handles or cross-hoods (see classification of surfaces ). ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \omega }$ ${\displaystyle \omega }$

If one leaves out the second countability in the definition of the manifold, then there are significantly more -bounded 2-manifolds. The Bagpipe Theorem (on German about bagpipes -Satz) by Peter Nyikos states that any such diversity still arises from a finite number of steps from a sphere by Henkel, cross-caps or so-called long pipes (English long pipes) from sticking, which with imagination a resemblance to a bagpipe emerges. We will not give the exact definition of a long pipe here. ${\displaystyle \omega }$

literature

• David Gauld: Nonmetrisable Manifolds. Springer, Auckland 2014

• Peter Nyikos: The theory of nonmetrizable manifolds. in Kunen, K., Vaughan, JE (eds.) Handbook of Set-Theoretic Topology , pp. 634-684, Amsterdam 1984

Individual evidence

1. ↑ Gauld: Nonmetrisable Manifolds. 2014, definition 4.6.
2. ↑ Gauld: Nonmetrisable Manifolds. 2014, Lemma 4.9.
3. ↑ Gauld: Nonmetrisable Manifolds. 2014, Corollary 4.8.
4. ↑ Engelking: General Topology. 1989, Example 3.10.19.
5. ↑ Gauld: Nonmetrisable Manifolds. 2014, Theorem A.49.
6. ↑ Gauld: Nonmetrisable Manifolds. 2014, Theorem 4.18.