In mathematics, an ω-bounded space is a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
in which the
closure of every
countable subset is
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
. More generally, if ''P'' is some property of subspaces, then a ''P''-bounded space is one in which every subspace with property ''P'' has compact closure.
Every compact space is ω-bounded, and every ω-bounded space is
countably compact In mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
Equivalent definitions
A topological space ''X'' is called countably compact if it satisfies any of the following equivalent condit ...
. The
long line Long line or longline may refer to:
*'' Long Line'', an album by Peter Wolf
* Long line (topology), or Alexandroff line, a topological space
*Long line (telecommunications), a transmission line in a long-distance communications network
*Longline fi ...
is ω-bounded but not compact.
The
bagpipe theorem
In mathematics, the bagpipe theorem of describes the structure of the connected (but possibly non-paracompact space, paracompact) ω-bounded space, ω-bounded surfaces by showing that they are "bagpipes": the connected sum of a compact space, co ...
describes the ω-bounded surfaces.
References
*
Properties of topological spaces
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