I am currently trying to understand Cech cohomology. Five questions arised and I would be glad for help. In what follows $X$ is a topological space.


I really like Dugger's and Isaksen's paper "Topological Hypercovers and ...". They prove for an arbitrary open cover $U=(U_a)_{a\in A}$ of $X$ the weak equivalence
$$
\operatorname{hocolim} ~C(U)\to X
$$
with $C(U)$ is the usual simplicial space of Cech namely
$$
\dots\to\coprod_{(b,c)\in A\times A}U_b\cap U_c\to\coprod_{a\in A}U_a
$$
Related statements are due to Segal. If $U$ is now a cover with every iterated intersection contractible we have a weak equivalence
$$
\operatorname{hocolim} ~C(U)\to \operatorname{hocolim} ~\pi_0C(U)
$$
The space $\pi_0C(U)$ is a simplicial set. You can find such a cover for every locally contractible space for instance. Dugger and Isaksen build the category $OpCov(X)$. This has open covers of $X$ as objects and morphisms are mappings between the index sets $f:A\to B$ and for every $a$ in $A$ a mapping $U_a\to V_{f(a)}$. One can also build the category $Cov(X)$ with same objects but morphisms only strict containments of covers. Is then
$$
\operatorname{hocolim} ~C(U)\sim \lim_{U\in Cov(X)}~\operatorname{hocolim} ~\pi_0C(U)\sim \lim_{U\in OpCov(X)}~\operatorname{hocolim} ~\pi_0C(U)
$$
all weakly equivalent for locally contractible $X$?
Why does one define Cech cohomology as $H^n(X)=\operatorname{colim} H^n(\operatorname{hocolim} C(U))$ with colimit over $Cov(X)^{op}$? Is this the right definition? Why not the limit over $Cov(X)$ instead of colimit over the opposite category?
What is the problem with Cech Homology? One defines $\hat H^n(X)=\operatorname{colim} ~H^n(C(U))$ as Cech cohomology. Then for a locally contractible $X$ it coincides with singular cohomology. Why not define $\hat H_n(X)=\lim ~H_n(C(U))$. I have read that it is because the limit functor does not respect exact sequences. Does it mean $\hat H_n(X)$ coincides not with singular homology for locally contractible $X$?
Can one see directly without going through sheaf cohomology that singular cohomology and Cech cohomology are the same for locally contractible $X$? I have written down complexes for the boundary of the two simplex with a nice cover and yes the cohomology is the same. Is there a kind of MayerVietoris argument?
Here is my last question. $X$ is a good space now. Where is the mistake in the following equality. I consider Top as a topologgical model category and the crucial step is perhaps the one which looks like a smallness argument on the homotopy category.
$$
\begin{array}{rcl}
\pi_0(X)&=&[S^0, \operatorname{hocolim} ~C(U)]\\
&=& \operatorname{hocolim}[S^0, C(U)]\\
&=& \operatorname{hocolim} \pi_0(C(U))\\&=&X
\end{array}
$$
Here $U$ is a cover with everything contractible as above.


Thank endurance reading and for help.

Newman gave an example in 1976 of a non-constant entire function bounded on each line through the origin in "An entire function bounded in every direction".  

I like the second sentence of the article:


  This is exactly what is needed to confuse students who have just struggled to comprehend the meaning of Liouville's theorem.


Armitage gave examples in 2007 of non-constant entire functions that go to zero in every direction in "Entire functions that tend to zero on every line".  For this I have only seen the MR review.  (If you don't have MathSciNet access, the link should still give you the publication information to find the article.)



Update: I just decided to take a look at the Armitage paper, and the introduction was enlightening:


  Although every bounded entire (holomorphic) function on $\mathbb{C}$
  is constant (Liouville’s theorem), it has been known for more than a hundred years
  that there exist nonconstant entire functions $f$ such that $f(z) → 0$ as $z →∞$ along
  every line through 0 (see, for example, Lindelöf’s book [10, pp. 119–122] of 1905). And it has been known for more than eighty years that such functions can tend to 0
  along any line whatsoever (see Mittag-Leffler [11], Grandjot [8], and Bohr [4]). Further
  references to related work are given in Burckel’s review [5] of Newman’s note [12].
  Entire functions with radial decay are used by Beardon and Minda [3] and Ullrich [14]
  in studies of pointwise convergent sequences of entire functions.


Armitage goes on to mention that Mittag-Leffler and Grandjot also gave explicit constructions, but states, "The examples given in what follows may nevertheless
be of some interest because of their comparative simplicity."  The examples are 
$$F(z)=\exp\left(-\int_0^\infty t^{-t}\cosh(tz^2)dt\right) - \exp\left(-\int_0^\infty t^{-t}\cosh(2tz^2)dt\right)$$ and
$$G(z)=\int_0^\infty e^{i\pi t}t^{-t}\cosh(t\sqrt{z})dt\int_0^\infty e^{i\pi t}t^{-t}\cos(t\sqrt{z})dt .$$
