Lindelof number

The Lindelöf number is one of the cardinals that characterizes the topological space . Defined as smallest cardinal $m$ ${\ displaystyle m}$ $m$ such that from each open covering of space $X$ ${\ displaystyle X}$ $X$ you can choose a sub-coverage of power not more $m$ ${\ displaystyle m}$ $m$ ^[1] . Designated as $l(X)$ ${\ displaystyle l (X)}$ ${\ displaystyle l (X)}$ . Since even a finite subcovering can be chosen in compact sets, the Lindelöf number in finite cases is taken as $\aleph _{0}$ ${\ displaystyle \ aleph _ {0}}$ $\ aleph_0$ (final cases, as a rule, are of no interest). If the Lindelof number of the space $X$ ${\ displaystyle X}$ $X$ equally $\aleph _{0}$ ${\ displaystyle \ aleph _ {0}}$ $\ aleph_0$ then $X$ ${\ displaystyle X}$ $X$ called lindelof space .

Properties

Lindelof number of space $X$ ${\ displaystyle X}$ $X$ no higher than network weight $X$ ${\ displaystyle X}$ $X$ $(l(X)\leqslant nw(X))$ ${\ displaystyle (l (X) \ leqslant nw (X))}$ $(l(X)\leqslant nw(X))$ ^[one]
Hausdorff space power $X$ ${\ displaystyle X}$ $X$ not more than $2^{l(X)*\chi (X)}$ ${\ displaystyle 2 ^ {l (X) * \ chi (X)}}$ $2^{l(X)*\chi (X)}$ where $\chi (X)$ ${\ displaystyle \ chi (X)}$ $\chi (X)$ - the nature of the topological space $X$ ${\ displaystyle X}$ $X$ ^[2]

Examples

$l(\mathbb {R} ^{n})=\aleph _{0}$ ${\ displaystyle l (\ mathbb {R} ^ {n}) = \ aleph _ {0}}$ $l(\mathbb {R} ^{n})=\aleph _{0}$
$l(L)=2^{\aleph _{0}}$ ${\ displaystyle l (L) = 2 ^ {\ aleph _ {0}}}$ $l(L)=2^{\aleph _{0}}$ where $L$ ${\ displaystyle L}$ $L$ - Nemytsky plane
$l(J(m))=m$ ${\ displaystyle l (J (m)) = m}$ $l(J(m))=m$ where $J(m)$ ${\ displaystyle J (m)}$ $J(m)$ - prickly hedgehog $m$ ${\ displaystyle m}$ $m$
The Lindelöf number of the direct Sorgenfrey is countably
The Lindelöf number of the square of the Sorgenfrey line is equal to the continuum

Notes

↑ ¹ ² Engelking, 1986 , p. 293.
↑ Engelking, 1986 , p. 342.

Literature

Engelking, Ryszard. General topology. - M .: Mir , 1986 .-- S. 290-293. - 752 s.

[_c419dbf5a03212db-1] ¹ ² Engelking, 1986 , p. 293.

[_c4166af5a02f1d3c-2] Engelking, 1986 , p. 342.

Lindelof number

Properties

Examples

Notes

Literature

More articles: