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==Homology==
A topological space  $  M $
with a unique non-trivial reduced homology group:

$$ 
\widetilde{H}  _ {k} ( M)  =  G ; \ \ 
\widetilde{H}  _ {i} ( M)  =  0 ,\ \ 
i \neq k .
$$

If  $  K ( \mathbf Z , n ) $
is the [[Eilenberg–MacLane space|Eilenberg–MacLane space]] of the group of integers  $  \mathbf Z $
and  $  M _ {k} ( G) $
is the Moore space with  $  \widetilde{H}  _ {k} ( M _ {k} ( G) ) = G $, 
then

$$ 
\lim\limits _ {N \rightarrow \infty } \ 
\left [
\Sigma  ^ {N+} k X , K ( \mathbf Z , N + n ) \wedge M _ {k} ( G) \right ]
 \cong  H  ^ {n} ( X , G ) ,
$$

that is,  $  \{ K ( \mathbf Z , n ) \wedge M _ {k} ( G) \} $
is the spectrum of the cohomology theory  $  H  ^ {*} (  , G ) $. 
This allows one to extend the idea of cohomology with arbitrary coefficients to a generalized cohomology theory. For any spectrum  $  E $, 
the spectrum  $  E \wedge M _ {k} ( G) $
defines a cohomology theory  $  ( E \wedge M _ {k} ( G) )  ^ {*} $, 
called the  $  E  ^ {*} $-
cohomology theory with coefficient group  $  G $. 
For the definition of generalized homology theories with coefficients in a group  $  G $, 
the so-called co-Moore space  $  M  ^ {k} ( G) $
is used, which is characterized by

$$ 
\widetilde{H}  {}  ^ {k} ( M  ^ {k} ( G) )  =  G ,\ \ 
\widetilde{H}  {}  ^ {i} ( M  ^ {k} ( G) )  =  0 ,\ \ 
i \neq k .
$$

For example, the group  $  \pi _ {i} ( X , G ) = [ M  ^ {k} ( G) , X ] $
is called the homotopy group of the space  $  X $
with coefficients in  $  G $. 
However, the space  $  M  ^ {k} ( G) $
does not exist for all pairs  $  ( G , k ) $. 
If  $  G $
is a finitely-generated group, then  $  M  ^ {k} ( G) $
does exist.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J.C. Moore,   "On homotopy groups of spaces with a single non-vanishing homotopy group"  ''Ann. of Math.'' , '''59''' :  3  (1954)  pp. 549–557</TD></TR></table>

====Comments====
For a construction of a Moore space as a CW-complex with one zero cell and further only cells in dimensions  $  n $
and  $  n + 1 $, 
cf. [[#References|[a1]]]. The Eilenberg–MacLane space  $  K ( G , n ) $
can be obtained from the Moore space  $  M ( G , n ) $
by killing the higher homotopy groups.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Gray,   "Homotopy theory. An introduction to algebraic topology" , Acad. Press  (1975)  pp. §17</TD></TR>
</table>

==General topology==
In general topology, a Moore space is a [[regular space]] with a [[Refinement|development]]: a sequence  $  \{ {\mathcal U} _ {n} \} _ {n} $
of open coverings such that for every  $  x $
and every open set  $  O $
containing  $  x $
there is an  $  n $
such that
$$ 
 \mathop{\rm St} ( x , {\mathcal U} _ {n} )  =  \cup
\{ {U \in {\mathcal U} _ {n} } : {x \in U } \}
 \subseteq  O ;
$$
(in other words,  $  \{  \mathop{\rm St} ( x , {\mathcal U} _ {x} ) \} _ {n} $
is a [[neighbourhood base]] at  $  x $.)

The idea of a development can be found in [[#References|[a4]]] (Axiom 1). Moore spaces are generalizations of [[metric space]]s and one can show that [[Collection-wise normal space|collectionwise normal]] Moore spaces are metrizable [[#References|[a2]]]. The question whether every normal Moore space is metrizable generated lots of research; its solution is described in [[#References|[a3]]].

====References====
<table>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  R.H. Bing,   "Metrization of topological spaces"  ''Canad. J. Math.'' , '''3'''  (1951)  pp. 175–186</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  W.G. Fleissner,   "The normal Moore space conjecture and large cardinals"  K. Kunen (ed.)  J.E. Vaughan (ed.) , ''Handbook of Set-Theoretic Topology'' , North-Holland  (1984)  pp. 733–760</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R.L. Moore,   "Foundations of point set theory" , Amer. Math. Soc.  (1962)</TD></TR>
</table>