Universal space

Universal space (relative to some class of topological spaces ${\mathcal {K}}$ ${\ displaystyle {\ mathcal {K}}}$ $\ mathcal {K}$ ) - topological space $X$ ${\ displaystyle X}$ $X$ such that $X$ ${\ displaystyle X}$ $X$ belongs to class ${\mathcal {K}}$ ${\ displaystyle {\ mathcal {K}}}$ $\ mathcal {K}$ and every space $Y$ ${\ displaystyle Y}$ $Y$ from the class ${\mathcal {K}}$ ${\ displaystyle {\ mathcal {K}}}$ $\ mathcal {K}$ invested in $X$ ${\ displaystyle X}$ $X$ , i.e $Y$ ${\ displaystyle Y}$ $Y$ homeomorphic to the subspace of space $X$ ${\ displaystyle X}$ $X$ . Using universal spaces, one can reduce the study of the class of topological spaces to the study of subspaces of a particular space ^[1] . Often, to prove the universality of space, the diagonal mapping theorem ^[1] ^{[2] is used} .

Content

1 Examples
- 1.1 Examples of universal spaces (hereinafter ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ $\ mathfrak {m}$ - a cardinal such that ${\mathfrak {m}}\geqslant \aleph _{0}$ ${\ displaystyle {\ mathfrak {m}} \ geqslant \ aleph _ {0}}$ $\ mathfrak {m} \ geqslant \ aleph_0$ , i.e ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ $\ mathfrak {m}$ endless):
2 notes
3 Literature

Examples

Examples of universal spaces (hereinafter ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ - a cardinal such that ${\mathfrak {m}}\geqslant \aleph _{0}$ ${\ displaystyle {\ mathfrak {m}} \ geqslant \ aleph _ {0}}$ , i.e ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ infinite ):

Alexander cube $F^{\mathfrak {m}}$ ${\ displaystyle F ^ {\ mathfrak {m}}}$ - ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ connected colon degree $F$ ${\ displaystyle F}$ (i.e. spaces $\{0;1\}$ ${\ displaystyle \ {0; 1 \}}$ with a topology consisting of an empty set , all space and a set $\{0\}$ ${\ displaystyle \ {0 \}}$ ) Is universal for all T ₀ -spaces of weight ${\mathfrak {m}}\geqslant \aleph _{0}$ ${\ displaystyle {\ mathfrak {m}} \ geqslant \ aleph _ {0}}$ ^[3] .
Tikhonovsky cube $I^{\mathfrak {m}}$ ${\ displaystyle I ^ {\ mathfrak {m}}}$ - ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ 1st degree of a single segment $I=[0;1]$ ${\ displaystyle I = [0; 1]}$ - universal for all Tikhonov spaces of weight ${\mathfrak {m}}\geqslant \aleph _{0}$ ${\ displaystyle {\ mathfrak {m}} \ geqslant \ aleph _ {0}}$ and for all compact Hausdorff spaces of weight ${\mathfrak {m}}\geqslant \aleph _{0}$ ${\ displaystyle {\ mathfrak {m}} \ geqslant \ aleph _ {0}}$ ^[4] .
Hilbert Cube $I^{\aleph _{0}}$ ${\ displaystyle I ^ {\ aleph _ {0}}}$ - the countable degree of a unit segment is universal for all metrizable compact spaces and for all metrizable separable spaces ^[5] .
$J({\mathfrak {m}})^{\aleph _{0}}$ ${\ displaystyle J ({\ mathfrak {m}}) ^ {\ aleph _ {0}}}$ - estimated degree of hedgehog prickly ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ - universal for all metrizable spaces of weight ${\mathfrak {m}}\geqslant \aleph _{0}$ ${\ displaystyle {\ mathfrak {m}} \ geqslant \ aleph _ {0}}$ ^[6] .
Space of rational numbers $\mathbb {Q}$ ${\ displaystyle \ mathbb {Q}}$ (with natural topology) is universal for all countable metrizable spaces ^[7] .
Cantor cube $D^{\mathfrak {m}}$ ${\ displaystyle D ^ {\ mathfrak {m}}}$ - ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ -th power of a two-point discrete space - universal for all zero-dimensional spaces of weight ${\mathfrak {m}}\geqslant \aleph _{0}$ ${\ displaystyle {\ mathfrak {m}} \ geqslant \ aleph _ {0}}$ ^[8] .
Baire's space $B({\mathfrak {m}})=D({\mathfrak {m}})^{\aleph _{0}}$ ${\ displaystyle B ({\ mathfrak {m}}) = D ({\ mathfrak {m}}) ^ {\ aleph _ {0}}}$ - calculated degree of discrete power space ${\mathfrak {m}}$ ${\ displaystyle {\ mathfrak {m}}}$ - universally for all metrizable spaces of weight zero in the sense of Ind ${\mathfrak {m}}\geqslant \aleph _{0}$ ${\ displaystyle {\ mathfrak {m}} \ geqslant \ aleph _ {0}}$ ^[9] .
Subspace of euclidean space $\mathbb {R} ^{2n+1}$ ${\ displaystyle \ mathbb {R} ^ {2n + 1}}$ formed by all points, no more than $n$ ${\ displaystyle n}$ whose coordinates are rational, universally for all metrizable separable spaces of dimension no more $n$ ${\ displaystyle n}$ ^[10] .
There is a compact, universal for all Tikhonov spaces $X$ ${\ displaystyle X}$ weights ${\mathfrak {m}}\geqslant \aleph _{0}$ ${\ displaystyle {\ mathfrak {m}} \ geqslant \ aleph _ {0}}$ such that $\dim X\leqslant n$ ${\ displaystyle \ dim X \ leqslant n}$ (i.e. Lebesgue dimension $X$ ${\ displaystyle X}$ not more $n$ ${\ displaystyle n}$ ) ^[11] .

Notes

↑ ¹ ² Engelking, 1986 , p. 136-137.
↑ Kelly, 1968 , p. 157-159.
↑ Engelking, 1986 , p.138.
↑ Engelking, 1986 , p. 137.
↑ Engelking, 1986 , p. 387.
↑ Engelking, 1986 , p. 418.
↑ Engelking, 1986 , p. 413.
↑ Engelking, 1986 , p. 544.
↑ Engelking, 1986 , p. 596.
↑ Engelking, 1986 , p. 618.
↑ Engelking, 1986 , p. 617.

Literature

Engelking, R. General Topology. - M .: Mir , 1986 .-- 752 p.
Kelly, J.L. General Topology. - M .: Science, 1968.

[_c53d02e143cf90f5-1] ¹ ² Engelking, 1986 , p. 136-137.

[_d5661596584e277b-2] Kelly, 1968 , p. 157-159.

[_81d3c930f2fea91b-3] Engelking, 1986 , p.138.

[_8c2f4430f956892e-4] Engelking, 1986 , p. 137.

[_89b7c63cd8019df3-5] Engelking, 1986 , p. 387.

[_ac0dd904c88a358a-6] Engelking, 1986 , p. 418.

[_0de2a20500ac85a3-7] Engelking, 1986 , p. 413.

[_0a88b8cfc18f3acb-8] Engelking, 1986 , p. 544.

[_5ba66723297895cb-9] Engelking, 1986 , p. 596.

[_d56899a6e053d3e0-10] Engelking, 1986 , p. 618.

[_2f97e2a711f707d9-11] Engelking, 1986 , p. 617.