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As an operator algebraist, I think the space of continuous functions of compact support to $\mathbb{C}$ which vanish at infinity is my preferred choice. Let me tell you why.

One of the basic ideas of noncommutative topology/geometry (and probably algebraic geometry, but I don't know much about that) is that we can trade the space for algebras of functions on that space. This is afforded by the Gelfand transform. The spectrum of a commutative $C^\ast$-algebra is the space of characters, i.e., $\ast$-algebra homomorphisms to $\mathbb{C}$.

• If $X$ is compact Hausdorff, then the spectrum of $C(X)$ is $X$.
• If $X$ is locally compact Hausdorff, but not compact, the spectrum of the non-unital $C^\ast$-algebra $C_0(X)$ is $X$. The spectrum of the unitalization of $C_0(X)$ ($C_0(X)\oplus \mathbb{C}$) is the one point compactification of $X$. The spectrum of the unital $C^\ast$-algebra $C_b(X)$ is $\beta X$, the Stone-Cech compactification of $X$. One should note that $C_b(X)$ is also the multiplier algebra of $C_0(X)$.
• If $X$ is compact, but not Hausdorff, then $C(X)$ corresponds to some type of "Hausdorffization" of $X$.

Actually $C(X)$ and $C_0(X)$ are the same if $X$ is compact, but you want to denote it $C(X)$ to emphasize the fact that the algebra is already unital. Otherwise, when you add a unit, you take the one point compactification of a compact space which adds an extra point, which is not what you want.

Now let's suppose you have some additional structure, like $X$ is a compact manifold. Then you probably want the $C^\infty$-functions on $X$. However, these can be recovered from $C(X)$ as those operators whose iterated commutator with the Dirac operator is bounded. This inspired the notion of a spectral triple.

EDIT: In my haste to answer this question, I made some mistakes in the earlier answer as pointed out by @Jonas.

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As an operator algebraist, I think $C(X)$, the space of continuous functions of compact support to $\mathbb{C}$, \mathbb{C}$is my preferred choice. Let me tell you why. One of the basic ideas of noncommutative topology/geometry (and probably algebraic geometry, but I don't know much about that) is that we can trade the space for algebras of functions on that space. This is afforded by the Gelfand transform. The spectrum of a commutative$C^\ast$-algebra is the space of characters, i.e.,$\ast$-algebra homomorphisms to$\mathbb{C}$. • If$X$is compact Hausdorff, then the spectrum of$C(X)$is$X$. The spectrum of$C_b(X)$is$\beta X$, the Stone-Cech compactification of$X$. One should note that$C_b(X)$is also the multiplier algebra of$C(X)$, so I would take$C(X)$to be a better choice. • If$X$is locally compact Hausdorff, but not compact, the spectrum of the non-unital$C^\ast$-algebra$C_0(X)$is$X$. The spectrum of unitalization of$C(X)$then corresponds to C_0(X)$ ($C_0(X)\oplus \mathbb{C}$) is the one point compactification of $X$. The spectrum of the unital $C^\ast$-algebra $C_b(X)$ is $\beta X$, the Stone-Cech compactification of $X$. One should note that $C_b(X)$ is also the multiplier algebra of $C_0(X)$.
• If $X$ is compact, but not Hausdorff, then $C(X)$ corresponds to some type of "Hausdorffization" of $X$.

Actually $C(X)$ and $C_0(X)$ are the same if $X$ is compact, but you want to denote it $C(X)$ to emphasize the fact that the algebra is already unital. Otherwise, when you add a unit, you take the one point compactification of a compact space which adds an extra point, which is not what you want.

Now let's suppose you have some additional structure, like $X$ is a compact manifold. Then you probably want the $C^\infty$-functions on $X$. However, these can be recovered from $C(X)$ as those operators whose iterated commutator with the Dirac operator is bounded. This inspired the notion of a spectral triple.

EDIT: In my haste to answer this question, I made some mistakes in the earlier answer as pointed out by @Jonas.

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As an operator algebraist, I think $C(X)$ C(X)$, the space of continuous functions to$\mathbb{C}$, is my preferred choice. Let me tell you why. One of the basic ideas of noncommutative topology/geometry (and probably algebraic geometry, but I don't know much about that) is that we can trade the space for algebras of functions on that space. This is afforded by the Gelfand transform. The spectrum of a commutative$C^\ast$-algebra is the space of characters, i.e.,$\ast$-algebra homomorphisms to$\mathbb{C}$. • If$X$is compact Hausdorff, then the spectrum of$C(X)$is$X$. The spectrum of$C_b(X)$is$\beta X$, the Stone-Cech compactification of$X$. One should note that$C_b(X)$is also the multiplier algebra of$C(X)$, so I would take$C(X)$to be a better choice. • If$X$is locally compact Hausdorff, but not compact, the spectrum of the non-unital$C^\ast$-algebra$C_0(X)$is$X$.$C(X)$then corresponds to the one point compactification of$X$. • If$X$is compact, but not Hausdorff, then$C(X)$corresponds to some type of "Hausdorffization" of$X$. Actually$C(X)$and$C_0(X)$are the same if$X$is compact, but you want to denote it$C(X)$to emphasize the fact that the algebra is already unital. Otherwise, when you add a unit, you take the one point compactification of a compact space which adds an extra point, which is not what you want. Now let's suppose you have some additional structure, like$X$is a compact manifold. Then you probably want the$C^\infty$-functions on$X$. However, these can be recovered from$C(X)\$ as those operators whose iterated commutator with the Dirac operator is bounded. This inspired the notion of a spectral triple.

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