Return to Answer

• $\mathcal{S}$-regular if for every neighborhood $U$ of a point $p\in X$ there is a $f\in\mathcal{S}$ such that $f(p)=1$ and $f\equiv 0$ in $X\smallsetminus U$, i.e. $X$ has enough bump functions in $\mathcal{S}$ (this should actually be called $\mathcal{S}$-completely regular or $\mathcal{S}$-Tychonoff, but it is called that way in Kriegl-Michor, so whatever);

• $\mathcal{S}$-normal if given two disjoint closed subsets $A_0,A_1\subset X$ there is a $f\in\mathcal{S}$ such that $f|_{A_i}=i$, $i=0,1$, i.e. it satisfies the Urysohn lemma (this, I believe, should be the same definition as the one in Gelfand-Raikov-Shilov's book pointed by Yemon Choi's comment above). Equivalently (Theorem 16.2 (1) $\Leftrightarrow$ (3), pp. 165-166 of Kriegl-Michor), every locally finite open covering of $X$ has a partition of unity subordinate to it comprising only of elements of $\mathcal{S}$;

• $\mathcal{S}$-paracompact if every open covering of $X$ has a partition of unity subordinated to it comprising only of elements of $\mathcal{S}$. Equivalently (Theorem 16.2, pp. 165-166 of Kriegl-Michor), $X$ is paracompact and $\mathcal{S}$-normal.

• $\mathcal{S}$-regular if for every neighborhood $U$ of a point $p\in X$ there is a $f\in\mathcal{S}$ such that $f(p)=1$ and $f\equiv 0$ in $X\smallsetminus U$, i.e. $X$ has enough bump functions in $\mathcal{S}$ (this should actually be called $\mathcal{S}$-completely regular or $\mathcal{S}$-Tychonoff, but it is called that way in Kriegl-Michor, so whatever);

• $\mathcal{S}$-normal if given two disjoint closed subsets $A_0,A_1\subset X$ there is a $f\in\mathcal{S}$ such that $f|_{A_i}=i$, $i=0,1$, i.e. it satisfies the Urysohn lemma with elements of $\mathcal{S}$ (this, I believe, should be the same definition as the one in Gelfand-Raikov-Shilov's book pointed by Yemon Choi's comment above). Equivalently (Theorem 16.2 (1) $\Leftrightarrow$ (3), pp. 165-166 of Kriegl-Michor), every locally finite open covering of $X$ has a partition of unity subordinate to it comprising only of elements of $\mathcal{S}$;

• $\mathcal{S}$-paracompact if every open covering of $X$ has a partition of unity subordinated to it comprising only of elements of $\mathcal{S}$. Equivalently (Theorem 16.2, pp. 165-166 of Kriegl-Michor), $X$ is paracompact and $\mathcal{S}$-normal.

There are a few more characterizations of $\mathcal{S}$-normality in Theorem 16.2, pp. 165-167 of Kriegl-Michor besides the one cited above. R. Bonic and J. Frampton showed (Smooth Functions on Banach Manifolds, J. Math. Mech. 15 (1966) 877-898, see also Theorem 14.8, pp. 155 in Kriegl-Michor) that $X$ is $\mathcal{S}$-regular iff the topology of $X$ is initial with respect to $\mathcal{S}$, i.e. the inverse images of open subsets of $\mathbb{R}$ under elements of $\mathcal{S}$ form a subbasis for the topology of $X$, and J.C. Wells showed (Differentiable Functions on Banach Spaces with Lipschitz Derivatives, J. Diff. Geom. 8 (1973) 135-152, see also Theorem 16.10, pp. 171-172 of Kriegl-Michor) that if $X$ is Lindelöf and $\mathcal{S}$-regular, then it is $\mathcal{S}$-paracompact. It was not known at the time Kriegl-Michor's book was published if paracompactness and $\mathcal{S}$-regularity entail $\mathcal{S}$-paracompactness.

There are a few more characterizations of $\mathcal{S}$-normality in Theorem 16.2, pp. 165-167 of Kriegl-Michor besides the one cited above. Moreover, under the assumption that $\mathcal{S}$ is local (i.e. if $f\in C(X,\mathbb{R})$ is such that there is an open covering $\mathscr{U}$ of $X$ such that for every $U\in\mathscr{U}$ there is $f_U\in\mathcal{S}$ such that $f|_U=f_U$, then $f\in\mathcal{S}$) and that $h\circ f\in\mathcal{S}$ for all $h\in C^\infty(\mathbb{R},\mathbb{R})$, $f\in\mathcal{S}$, we have the following results (Edit: the previous assumptions were missing in a previous version of this answer):

• R. Bonic and J. Frampton showed (Smooth Functions on Banach Manifolds, J. Math. Mech. 15 (1966) 877-898, see also Theorem 14.8, pp. 155 in Kriegl-Michor) that $X$ is $\mathcal{S}$-regular iff the topology of $X$ is initial with respect to $\mathcal{S}$, i.e. the inverse images of open subsets of $\mathbb{R}$ under elements of $\mathcal{S}$ form a subbasis for the topology of $X$;

• J.C. Wells showed (Differentiable Functions on Banach Spaces with Lipschitz Derivatives, J. Diff. Geom. 8 (1973) 135-152, see also Theorem 16.10, pp. 171-172 of Kriegl-Michor) that if $X$ is Lindelöf and $\mathcal{S}$-regular, then it is $\mathcal{S}$-paracompact (actually, for this result we do not need $\mathcal{S}$ to be local or closed under composition with any $h\in C^\infty(\mathbb{R},\mathbb{R})$, only that $h\circ f\in\mathcal{S}$ for all $f\in\mathcal{S}$ and all $h\in C^\infty(\mathbb{R},\mathbb{R})$ such that $h(\mathbb{R})\subset[0,1]$, $h(t)=0$ for $t\leq 0$ and $h(t)=1$ if $t\geq 1$);

• It was not known at the time Kriegl-Michor's book was published if paracompactness and $\mathcal{S}$-regularity entail $\mathcal{S}$-paracompactness, even with the above hypotheses.

I will leave to Yemon Choi discussing the answer from Gelfand-Raikov-Shilov's book, and restrict myself to more recent discussions on the matter...

There is an extensive discussion on abstract partitions of unity in Chapter III of the book of Andreas Kriegl and Peter Michor, The Convenient Setting of Global Analysis (AMS, 1997). There they define precisely what you want - the concepts of $\mathcal{S}$-regularity, $\mathcal{S}$-normality and $\mathcal{S}$-paracompactness. To wit, let $X$ be a Hausdorff topological space and $\mathcal{S}\subset C(X,\mathbb{R})$ be a subalgebra. Then $X$ is:

• $\mathcal{S}$-regular if for every neighborhood $U$ of a point $p\in X$ there is a $f\in\mathcal{S}$ such that $f(p)=1$ and $f\equiv 0$ in $X\smallsetminus U$, i.e. $X$ has enough bump functions in $\mathcal{S}$ (this should actually be called $\mathcal{S}$-completely regular or $\mathcal{S}$-Tychonoff, but it is called that way in Kriegl-Michor, so whatever);

• $\mathcal{S}$-normal if given two disjoint closed subsets $A_0,A_1\subset X$ there is a $f\in\mathcal{S}$ such that $f|_{A_i}=i$, $i=0,1$, i.e. it satisfies the Urysohn lemma. Equivalently (Theorem 16.2 (1) $\Leftrightarrow$ (3), pp. 165-166 of Kriegl-Michor), every locally finite open covering of $X$ has a partition of unity subordinate to it comprising only of elements of $\mathcal{S}$;

• $\mathcal{S}$-paracompact if every open covering of $X$ has a partition of unity subordinated to it comprising only of elements of $\mathcal{S}$. Equivalently (Theorem 16.2, pp. 165-166 of Kriegl-Michor), $X$ is paracompact and $\mathcal{S}$-normal.

There are a few more characterizations of $\mathcal{S}$-normality in Theorem 16.2, pp. 165-167 of Kriegl-Michor besides the one cited above. R. Bonic and J. Frampton showed (Smooth Functions on Banach Manifolds, J. Math. Mech. 15 (1966) 877-898, see also Theorem 14.8, pp. 155 in Kriegl-Michor) that $X$ is $\mathcal{S}$-regular iff the topology of $X$ is initial with respect to $\mathcal{S}$, i.e. the inverse images of open subsets of $\mathbb{R}$ under elements of $\mathcal{S}$ form a subbasis for the topology of $X$, and J.C. Wells showed (Differentiable Functions on Banach Spaces with Lipschitz Derivatives, J. Diff. Geom. 8 (1973) 135-152, see also Theorem 16.10, pp. 171-172 of Kriegl-Michor) that if $X$ is Lindelöf and $\mathcal{S}$-regular, then it is $\mathcal{S}$-paracompact. It was not known at the time Kriegl-Michor's book was published if paracompactness and $\mathcal{S}$-regularity entail $\mathcal{S}$-paracompactness.

As for paracompactness itself, if $X$ is also locally compact, there is a nice algebraic characterization of it by R. Bkouche, whose proof was later simplified by R.L. Finney and J. Rotman (Paracompactness of Locally Compact Spaces, Michigan Math. J. 17 (1970) 359-361): A locally compact Hausdorff topological space $X$ is paracompact iff the ideal $C_c(X,\mathbb{R})\subset C(X,\mathbb{R})$ of continuous functions with compact support is a projective $C(X,\mathbb{R})$-module. The beautiful and simple argument of Finney and Rotman carries over to smoothly paracompact manifolds, and presumably should be adaptable to general $\mathcal{S}$.

I will leave to Yemon Choi discussing the answer from Gelfand-Raikov-Shilov's book (Commutative Normed Rings, I suppose?), and restrict myself to more recent discussions on the matter...

There is an extensive discussion on abstract partitions of unity in Chapter III of the book of Andreas Kriegl and Peter Michor, The Convenient Setting of Global Analysis (AMS, 1997). There they define precisely what you want - the concepts of $\mathcal{S}$-regularity, $\mathcal{S}$-normality and $\mathcal{S}$-paracompactness. To wit, let $X$ be a Hausdorff topological space and $\mathcal{S}\subset C(X,\mathbb{R})$ be a subalgebra. Then $X$ is:

• $\mathcal{S}$-regular if for every neighborhood $U$ of a point $p\in X$ there is a $f\in\mathcal{S}$ such that $f(p)=1$ and $f\equiv 0$ in $X\smallsetminus U$, i.e. $X$ has enough bump functions in $\mathcal{S}$ (this should actually be called $\mathcal{S}$-completely regular or $\mathcal{S}$-Tychonoff, but it is called that way in Kriegl-Michor, so whatever);

• $\mathcal{S}$-normal if given two disjoint closed subsets $A_0,A_1\subset X$ there is a $f\in\mathcal{S}$ such that $f|_{A_i}=i$, $i=0,1$, i.e. it satisfies the Urysohn lemma (this, I believe, should be the same definition as the one in Gelfand-Raikov-Shilov's book pointed by Yemon Choi's comment above). Equivalently (Theorem 16.2 (1) $\Leftrightarrow$ (3), pp. 165-166 of Kriegl-Michor), every locally finite open covering of $X$ has a partition of unity subordinate to it comprising only of elements of $\mathcal{S}$;

• $\mathcal{S}$-paracompact if every open covering of $X$ has a partition of unity subordinated to it comprising only of elements of $\mathcal{S}$. Equivalently (Theorem 16.2, pp. 165-166 of Kriegl-Michor), $X$ is paracompact and $\mathcal{S}$-normal.

There are a few more characterizations of $\mathcal{S}$-normality in Theorem 16.2, pp. 165-167 of Kriegl-Michor besides the one cited above. R. Bonic and J. Frampton showed (Smooth Functions on Banach Manifolds, J. Math. Mech. 15 (1966) 877-898, see also Theorem 14.8, pp. 155 in Kriegl-Michor) that $X$ is $\mathcal{S}$-regular iff the topology of $X$ is initial with respect to $\mathcal{S}$, i.e. the inverse images of open subsets of $\mathbb{R}$ under elements of $\mathcal{S}$ form a subbasis for the topology of $X$, and J.C. Wells showed (Differentiable Functions on Banach Spaces with Lipschitz Derivatives, J. Diff. Geom. 8 (1973) 135-152, see also Theorem 16.10, pp. 171-172 of Kriegl-Michor) that if $X$ is Lindelöf and $\mathcal{S}$-regular, then it is $\mathcal{S}$-paracompact. It was not known at the time Kriegl-Michor's book was published if paracompactness and $\mathcal{S}$-regularity entail $\mathcal{S}$-paracompactness.

As for paracompactness itself, if $X$ is also locally compact, there is a nice algebraic characterization of it by R. Bkouche, whose proof was later simplified by R.L. Finney and J. Rotman (Paracompactness of Locally Compact Spaces, Michigan Math. J. 17 (1970) 359-361): A locally compact Hausdorff topological space $X$ is paracompact iff the ideal $C_c(X,\mathbb{R})\subset C(X,\mathbb{R})$ of continuous functions with compact support is a projective $C(X,\mathbb{R})$-module. The beautiful and simple argument of Finney and Rotman carries over to smoothly paracompact manifolds, and presumably should be adaptable to general $\mathcal{S}$.