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# The (co)tangent sheaf of a topological space

Let $$X$$ be a topological space (assume additional assumptions if needed) and denote by $$\mathcal O _X$$ its sheaf of $$\Bbbk$$-valued continuous functions where $$\Bbbk$$ is $$\mathbb{R}$$ or $$\mathbb{C}$$ with standard topology.

Then, as it is done in the differentiable setting or in algebraic geometry, one can define the following objects $$T_X:=\mathscr{Der}_\Bbbk (\mathcal O_X,\mathcal O_X)$$ the tangent sheaf, i.e. the sheaf of $$\Bbbk$$-linear derivations of $$\mathcal O_X$$ with values in $$\mathcal O_X$$ (on local sections, $$\Bbbk$$-linear maps $$D:\mathcal O_X(U)\to\mathcal O_X(U)$$ satisfying Leibniz: $$D(f\cdot g)=f\cdot Dg + g\cdot Df$$), and $$\Omega_X^1:=\mathcal I/\mathcal I^2$$

the sheaf of differentials, where $$\mathcal I$$ is the ideal sheaf of $$X$$ embedded diagonally $$\Delta:X\hookrightarrow X\times X$$ into $$X\times X$$ (i.e. $$\mathcal I(U)=$$ functions in $$\mathcal O_{X\times X}(U)$$ that are zero on every point of $$\Delta(X)\subset X\times X$$).

Well, what can be said about these two sheaves? Anything interesting at all?

Also, is there any relationship between $$T_X$$ and the "tangent microbundle" $$\tau_X$$ in case $$X$$ is a topological manifold?

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