MathOverflow

In a variational model for morphogenesis defined on $H^2(\Omega)$ with Neumann boundary conditions, the energy functional involves a local term
$$E_{\text{loc}}(\vec{C}) = (\|\vec{C}\|-1)^2,$$
where $\|\cdot\|$ denotes the pointwise Euclidean norm (the total energy is then $\int_\Omega E_{\text{loc}}\,d\Omega$)....

In a talk by A. Bondal (pointed out to me in a comment here) the following theorem is stated:
If $X=\mathbb{C}^n/\mathbb{Z}^{2n}$ (for $n\geqslant3$) is a general torus (i.e. such that $H^{p,p}(X)\cap H^{2p}(X,\mathbb{Z})=0$ for all $0<p<\dim X$) then the inclusion $D^\mathrm{b}(\mathsf{Coh}(X))\to D_{\mathsf{coh}}^\mathrm{b}(X)$ is not...

I would like to know sufficient conditions under which
the direct limit of topological spaces is locally "nice" (connected, path-connected, contractible),
direct limit commutes with the direct product in the category of topological spaces.

https://www.desmos.com/calculator/tln4wxdcda
My question is what are the chances that a randomly chosen point on the function will be defined or undefined, are those chances 50% - 50%