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Let's consider an equation $\phi\overline u = \overline b$, where $\phi$ is a linear mapping and $b$ is a vector in a linear space $V$. The solution space $U$ to this equation describes a generalization to the concept of plane. Now what are these kind of sets called in particular cases?

For example if $V$ have finite dimension we would call U a point if $\dim(U) = 0$, line if $\dim(U) = 1$ plane if $\dim(U)=2$ and hyperplane if $\operatorname{codim}(U) = 1$.

But are there names for the cases where $\dim(U)=3$ or even $\dim(U)=4$? What about $\operatorname{codim}(U) = 2$ or $\operatorname{codim}(U) = 0$?

Do we call them the same if $V$ is of infinite dimensions?

Do this type of space have a name in general? Wikipedia suggest that they're called flats at least if $V$ have finite dimensions. Is this a term that is used even if $V$ have infinite dimensions (and also $U$)?

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    I think you've covered the most common names already. I do not know any more names for specific dimensions or codimensions. Dimension 0 has been called a "point", though, in most literature that I've read. – 2017-01-09
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    They are subspaces. – 2017-01-09
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    @Simon They are not necessarily subspaces for $b\ne 0$. In general they are only affine subspaces. – 2017-01-09
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    Oh yes, thank you ! How silly of me ! I agree with Marc's answer below. – 2017-01-09

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The general name is affine subspace (of the domain of $\phi$, viewed as affine space rather than as vector space). Special cases are called (affine) point ($\dim=0$), affine line ($\dim=1$), affine plane ($\dim=2$), affine hyperplane ($\operatorname{codim}=1$). Otherwise one can always use "affine subspace of dimension$~d$", or "affine $d$-space" to specify the dimension.