  We performed muon spin rotation/relaxation measurements to investigate the
magnetic behavior of $T^{\prime}$-type Eu$_2$CuO$_4$ (ECO), which is the parent
compound of electron-doped cuprate superconductors, and the effects of
oxygen-reduction annealing on its magnetism. In as-sintered (AS) ECO, we
clarified the development of magnetic correlations upon cooling below
$T_\mathrm{N1}$ (= 265 K) as well as the coexistence of a dominant fluctuating
spin state and partially ordered spin state in the temperature range between
$\sim$150 K and $T_\mathrm{N1}$. Upon further cooling, uniform long-range
magnetic order was observed below $T_\mathrm{N2} = 110$ K, which is close to
the ordering temperature of 115 K in $T^{\prime}$-type La$_2$CuO$_4$ (LCO)
[Phys. Rev. B {\bf 82}, 180508(R) (2010)]. For oxygen-reduction-annealed ECO, a
similar ordering sequence with the same $T_\mathrm{N2}$ was observed but
without the partially ordered spin state. Therefore, the fluctuating spin state
over a wide temperature range and a $T_\mathrm{N2}$ less than the N{\'e}el
temperature ($T_\mathrm{N2} \approx T_\mathrm{N1}$) in $T$-type LCO are common
features of the $T^{\prime}$-type parent $R_2$CuO$_4$ ($R$CO, $R$: rare-earth
ion). The origin of the partially ordered spin state in AS ECO is discussed
from the viewpoint of chemical defect. Furthermore, we discuss the roles of
electron doping and repairing defect in the observed effect of annealing on the
magnetism of $T^{\prime}$-type $R$CO.
