Let $X$ be a subset of a (finite dimensional) Banach space $E$. Let's call a function $f\colon X\longrightarrow \mathbb{K}$ smooth (in whatever sense we are interested in) if there is an open neighbourhood $X\subseteq U\subseteq E$ of $X$ and a smooth function $\widetilde{f}\colon U\longrightarrow\mathbb{K}$ such that $\widetilde{f}|_{X}=f$. Then $X$ with the induced topology and the sheaf defined by $\mathcal{C}_X^\infty(V)=\left\{\text{Smooth functions }V\longrightarrow \mathbb{K}\big.\right\}$ forms a locally $\mathbb{K}$-ringed space.

Let us define an ugly manifold to be a locally $\mathbb{K}$-ringed space which is locally isomorphic to a locally ringed space of the form $(X,\mathcal{C}_X^\infty)$, where $X$ is any subset of a (finite dimensional) Banach space. A manifold is a locally ringed space, locally isomorphic to $(U,\mathcal{C}_U^\infty)$ for some open subset $U$ of a Banach space. A manifold with boundary is a locally $\mathbb{R}$-ringed space, locally isomorphic to $(X,\mathcal{C}_X^\infty)$, where $X=\left\{x\in E\ \middle|\ \lambda(x)\ge 0\right\}$, for $E$ a Banach space and $\lambda$ some continous linear form on $E$; and similarly a manifold with corners can be defined.

So my questions is: Which properties of open subsets, half spaces and corners spaces (if this is the correct word) are essential for the theory of manifolds to work. What breaks down for ugly manifolds? tangent spaces, differential forms, etc. come to mind here. Is there a class of subsets of (finite dimensional) Banach spaces, which lead to nice manifolds and yield a more flexible category? For example not like manifolds with boundary, which are not closed under products.

Note that I know almost nothing about analysis; in particular I haven't yet taken a course on real analysis or something like that. I've only read most of Torsten Wedhorn's book Manifolds, Sheaves and Cohomology since I am more an algebraically minded type of person, so I'd appreciate, if your answer was accessible to such.

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    $\begingroup$ For a topological space, "locally" normally means "in some open subset", so you need to define what's meant by "locally isomorphic to a locally ringed space [of functions on] any subset of a Banach space". Regardless of the definition, one obvious snag is, the concept of "locally isomorphic to" must be transitive if "local structure" is to be well-defined. "Any subset of a Banach space" is awfully lax. $\endgroup$ – Andrew D. Hwang Jan 13 '17 at 19:05
  • $\begingroup$ @AndrewD.Hwang I mean a locally ringed space $(M,\mathcal{O}_M)$ such that every point $p$ has an open neighbourhood $U$ such that $(U,\mathcal{O}_M|_U)$ is isomorphic to $(X,\mathcal{C}_X^\infty)$, where the latter locally ringed space is defined as above. $\endgroup$ – Jakob Werner Jan 16 '17 at 6:48
  • $\begingroup$ 1) Tangent spaces can be defined as for every locally ringed space. 2) Have you tried to construct fiber products in this category? $\endgroup$ – HeinrichD Apr 5 '17 at 9:45

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