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A [[Topological space|topological space]] containing a homeomorphic image of every topological space of a certain class. Examples are: 1)  $  C [ 0 , 1 ] $, 
cf. [[Banach space|Banach space]]; 2) the [[Hilbert cube|Hilbert cube]] and the [[Tikhonov cube|Tikhonov cube]]; 3) the Menger curve (cf. [[Line (curve)|Line (curve)]]); 4) the universal Milnor bundle (cf. [[Principal fibre bundle|Principal fibre bundle]]).

The existence of universal spaces allows the consideration of abstract objects as subobjects (in the categorical sense) of a more concrete one, and thus endows them with a greater wealth of  "intrinsic"  properties. On the other hand, it emphasizes the relations of  "parts of a whole" .

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.S. Aleksandrov,   "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.M. Cohn,   "Universal algebra" , Reidel  (1981)</TD></TR></table>

====Comments====
There is also a dual notion of universal space: Every space in a certain class is a continuous image of the space in question. E.g., the [[Cantor set|Cantor set]] is universal for the class of compact metric spaces, the unit interval is universal for the class of locally connected continua (see [[Peano curve|Peano curve]]), and the [[Pseudo-arc|pseudo-arc]] is universal for the class of snake-like continua (see [[Cube-like continuum|Cube-like continuum]]).

==Universal spaces in functional analysis.==
There are various notions of a universal space in functional analysis. A topological vector space  $  X _ {0} $
is universal for a class  $  {\mathcal X} $
of topological vector spaces if for every  $  X \in {\mathcal X} $
there is a closed subspace of  $  X _ {0} $
isomorphic to  $  X $. 
There is always a trivial universal space for any class  $  {\mathcal X} $, 
but whether there is a universal space in  $  {\mathcal X} $
itself or in a closely related class is a different matter. The following theorem holds, [[#References|[a2]]]: There is a universal separable  $  F $-space for the class of all  $  F ^ { * } $-spaces. Here, an  $  F $-space and an  $  F ^ { * } $-space are defined as follows.

An  $  F $-norm on a linear space  $  X $
is a mapping  $  \| \| $
from  $  X $
to the non-negative real numbers such that:

1)  $  \| x \| = 0 $
if and only if  $  x= 0 $;

2)  $  \| ax \| = \| x \| $
for all  $  a $, 
$  | a | = 1 $;

3)  $  \| x+ y \| \leq  \| x \| + \| y \| $;

4)  $  \| a _ {n} x \| \rightarrow 0 $
if  $  a _ {n} \rightarrow 0 $;

5)  $  \| ax _ {n} \| \rightarrow 0 $
if  $  x _ {n} \rightarrow 0 $;

6)  $  \| a _ {n} x _ {n} \| \rightarrow 0 $
if  $  a _ {n} \rightarrow 0 $, 
$  x _ {n} \rightarrow 0 $.

In conditions 4), 5), 6) the topology is the one induced by the translation-invariant metric  $  \rho ( x, y)= \| x- y \| $. 
If  $  \| \| $
is not required to satisfy 1), it is called an  $  F $-pseudo-norm. An  $  F $-norm or  $  F $-pseudo-norm is  $  p $-homogeneous if  $  \| ax \| = | a |  ^ {p} \| x \| $. 
A  $  1 $-homogeneous  $  F $-norm ( $  F $-pseudo-norm) is a [[Norm|norm]] ([[Pseudo-norm|pseudo-norm]]). An  $  F ^ { * } $-space is a linear space with an  $  F $-norm; an  $  F $-space is a complete  $  F ^ { * } $-space.

Some other universality results:  $  C[ 0, 1] $
is universal for all separable Banach spaces (the Banach–Mazur theorem, cf. [[Metric space|Metric space]]);  $  C( - \infty , \infty ) $
is universal for all separable  $  B _ {0} $-spaces (a  $  B _ {0} ^ { * } $-space is a locally convex metric linear space and a  $  B _ {0} $-space is a complete  $  B _ {0} ^ { * } $-space); there is a separable locally pseudo-convex space which is universal for all separable locally pseudo-convex spaces; there is a separable locally bounded complete space  $  X $
with a  $  p $-homogeneous norm which is universal for all separable locally bounded spaces with  $  p $-homogeneous norms. (Here, an  $  F $-space is locally bounded if it contains a bounded neighbourhood of  $  0 $; 
a locally pseudo-convex space is a metric linear space whose topology can be given by a family of  $  p _ {n} $-homogeneous pseudo-norms.)

The dual notion is that of a co-universal linear space. An  $  F $-space  $  X _ {0} $
is co-universal for a family  $  {\mathcal X} $
of  $  F $-spaces if every element of  $  {\mathcal X} $
is isomorphic to a quotient space  $  X _ {0} /Y $
for a closed subspace  $  Y $
of  $  X _ {0} $.

Some co-universality results: there is a separable  $  F $-space which is co-universal for all separable  $  F $-spaces, [[#References|[a2]]]; every separable locally bounded space  $  X $
with a  $  p $-homogeneous norm is an image of  $  l _ {p} $
under a continuous linear operator, [[#References|[a3]]]–[[#References|[a5]]].

Cf. [[#References|[a6]]] for a large number of universality and co-universality theorems for metric linear spaces, including all of the above.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Engelking,   "General topology" , Heldermann  (1989)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.J. Kalton,   "Universal spaces and universal bases in metric linear spaces"  ''Studia Math.'' , '''61'''  (1977)  pp. 161–191</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.H. Shapiro,   "Examples of proper closed weakly dense subspaces in non-locally convex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576079.png" />-spaces"  ''Isr. J. Math.'' , '''7'''  (1969)  pp. 369–380</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  W.J. Stiles,   "On properties of subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095760/u09576081.png" />"  ''Trans. Amer. Math. Soc.'' , '''149'''  (1970)  pp. 405–415</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Banach,   "Théorie des opérations linéaires" , PWN  (1932)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Rolewicz,   "Metric linear spaces" , Reidel  (1985)  pp. 44</TD></TR></table>