I may be asking something a little out of my comfort zone at this moment so bear with me.

Before I begin let me provide some background for the interested outsider:

Let $X$ be a topological space and fix some $x\in X$. We can consider the set $\mathcal{U}_x$ of all neighborhoods of $x$ in $X$. This is a directed set with the reverse inclusion ordering $\subseteq_{\text{op}}$. We can note then that if we consider the set $\{C(U,\mathbb{R})\}_{U\in\mathcal{U}_x}$ (where each $C(U,\mathbb{R})$ denotes, as usual, the set of continuous maps $U\to\mathbb{R}$ where we are thinking about it as a $\mathbb{R}$-space) and the set $\{\text{res}_{U,V}:C(U,\mathbb{R})\to C(V,\mathbb{R})\}_{U,V\in\mathcal{U}_x,\; U\subseteq_\text{op}V}$ (where $\text{res}_{U,V}:C(U,\mathbb{R})\to C(V,\mathbb{R})$ is the restriction map taking $C(U,\mathbb{R})\ni f\mapsto f_{\mid V}\in C(V,\mathbb{R})$) then we have a nice directed system of $\mathbb{R}$-spaces. We can then consider the direct limit $\varinjlim C(U;\mathbb{R})$ to get a $\mathbb{R}$-space. Call this space the germs of continuous functions at $x$ and denote it $\mathcal{G}_x$. It's not hard to prove (it's intuitively obvious from the direct limit definition) that $\mathcal{G}_x$ admits a different description as $C(X,\mathbb{R})/\sim$ where $f\sim g$ if and only if there exists $U\in\mathcal{U}_x$ with $f_{\mid U}=g_{\mid U}$ and we add/multiply scalars representative wise.

Anyways, so I completely understand (or at least I hope I do) the intuitive idea behind the germ of continuous functions. That said, I have a hard time actually computing it except for very extreme scenarios. For example, if $x$ is an isolated point of $X$ I see quite clearly that $\mathcal{G}_x\to\mathbb{R}:[f]\mapsto f(x)$ is a well-defined isomorphism. But, in general, I have no idea how to find nice descriptions of the germs of functions at a point. Is there any general methodology in finding them, or some general class of examples that admit a nice description?

Thanks for your time!

  • $\begingroup$ If there were a simple, intuitive, description, why would all the books use that complicated one? They wouldn't. $\endgroup$
    – GEdgar
    Commented Nov 30, 2011 at 1:16
  • $\begingroup$ What do you want to compute? The set of germs at a point? Try doing the case of analytic functions, if you want something with a reasonable description. $\endgroup$ Commented Nov 30, 2011 at 2:06
  • $\begingroup$ @GEdgar I disagree. If one intuitively understands what the direct limit means and what the directed system is measuring then it should be clear. Said differently, two things in a directed limit are intuitively equal if they are "eventually equal" which in this case is precisely the more intuitive definition. The first definition is just nice for technical reasons. $\endgroup$ Commented Nov 30, 2011 at 2:29
  • $\begingroup$ @MarianoSuárez-Alvarez I want to be able to actually find isomorphisms between specific examples of germs of functions and more common spaces. i.e. $\mathcal{G}_x\cong\mathbb{R},\mathbb{R}[x]$ For example, like I said, if $x$ is an isolated point then (I believe) $\mathcal{G}_x\cong\mathbb{R}$. My question is in the same vein as wanting to realize $\mathbb{C}[x]/(x^2+1)$ as $\mathbb{C}^2$--something complicated is isomorphic to something more manageable. If this isn't feasible, I would like to understand intuitively why. $\endgroup$ Commented Nov 30, 2011 at 2:31
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    $\begingroup$ Because there are waaaaay too many continuous functions. To give an example: suppose you wanted instead germs of $C^\infty$ functions $\mathbb R\to\mathbb R$ at $0$; there are many more such germs than possible Taylor series, and a theorem of Borel tells us that every formal power series is the Taylor series of a function. (In particular, what happens at an isolated point is not a good indication of what to expect even in the slightly more general case of germs of real functions defined on $\mathbb R$) $\endgroup$ Commented Nov 30, 2011 at 3:43


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