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?