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I learned that topological manifolds as Hausdorff, second-countable topological spaces that are locally homeomorphic to $\mathbb{R}^n$. And then smooth manifolds (or also $C^r$ and real-analytic manifolds) are topological manifolds with a smooth atlas. I know there are also complex manifolds. However, for me it seems like these definitions are very dependent upon $\mathbb{R}^n$ and $\mathbb{C}^n$.

Is there a more general definition for a topological manifold or smooth manifold, but something that is more specific than just a topological space--maybe something that doesn't make reference to $\mathbb{R}$ or $\mathbb{C}$? Maybe it has some notions that make sense that wouldn't make sense for a general topological space such as tangent vectors (in the case of smooth manifolds).

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    $\begingroup$ If you want something beyond a topological manifold, you need some vector space in which to do calculus, after all. There are also Banach manifolds and Fréchet manifolds ... $\endgroup$ Apr 2, 2017 at 21:51
  • $\begingroup$ You might be interested in topological vector spaces or even more generally uniform spaces. $\endgroup$
    – hardmath
    Sep 7, 2020 at 15:14

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Let me elaborate a bit my comment.

If $X$ is a topological space, a presheaf $F$ of rings is a data of rings $F(U)$ for all $U \subset X$ open, and of "restriction morphism" $F(U) \to F(V)$ if $U \supset V$. A sheaf is a presheaf with some conditions (gluing and uniqueness). This can be though as "generalized functions".

A ringed space is a topological space $X$ and a sheaf $\mathcal O_X$ of rings. This is really describing a lot of situations you already know (topological, smooth manifold). If your ringed space is a locally ringed space, then you can define tangent space for example. This construction is very general and used in scheme theory usually but I think if you google there are probably some articles or lectures notes which study general ringed spaces.

Let me give you the definition of the tangent space in this setting : as you have maybe seen, for a manifold a possible definition of $T_xM$ is $\text{Hom}(m_x/m_x^2,\mathbb R)$ where ${m_x = \{f \in C^{\infty}(M) : f(x) = 0\} }$. This definition, very algebraic perfectly extends to any locally ringed space, and if $\mathfrak m_x$ is the maximal ideal of $\mathcal O_{X,x}$ then $T_xX = \text{Hom}(\mathfrak m_x/ \mathfrak m_x^2,k_x)$ where $k_x$ is the residual field.

You can find more informations in these lectures notes which define a manifold via sheaves.

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