Let $f:X\times [0,1] \to Y$ and let $X$ be compact. There exist points $0= t_0 < t_1 < · · · < t_k = 1$ such that $F(a, t_{i-1}) $,$F(a, t_{i})\in U$

Question

Let $f:X\times [0,1] \rightarrow Y$ be a continuous function, let $X$ and $Y$ be compact Hausdporff, and let $\mathcal{U}$ be an open cover of $Y$. If $a\in X$, then there exist points $0= t_0 < t_1 < · · · < t_k = 1$ and an open subset $U\in \mathcal{U} $ of $Y$ such that $F(a, t_{i-1}) $,$F(a, t_{i})\in U$ for each $i$.

I need this result to prove a theorem about simplicial approximations of homotopic maps. Here it is:

Answer 1

I understand your problem to be:

compact X, f in C(Xx[0,1],Y), C open cover Y, a in X
implies some t0,.. t_k in [0,1], U in C with t0 = 0, t_k = 1
and for j = 1,.. k, f(a,t_(j-1)), f(a,t_j) in U.

Note that implies for j = 0,1,.. k, f(a,t_j) in U.
A counter example is X = {a}, Y = [0,1], C = { [0,1), (0,1] }

Could there be a quantification problem?
Is this a correct statement of your problem?

compact X, f in C(Xx[0,1],Y), C open cover Y, a in X
implies some t0,.. t_k in [0,1], with t0 = 0, t_k = 1 and
for j = 1,.. k, some U_j in C with f(a,t_(j-1)), f(a,t_j) in U.

Answer 2

compact X, f in C(Xx[0,1],Y), C open cover Y, a in X
implies some t0,.. t_k in [0,1], with t0 = 0, t_k = 1 and
for j = 1,.. k, some U_j in C with f(a,t_(j-1)), f(a,t_j) in U.

Proof. g = f restriced to {a}x[0,1] is continuous.
{ g^-1(U) | U in c } is open cover of {a}x[0,1].
For U in C let O_U = pi_2(g^-1(U)).
{ O_U : U in C } is an open cover of [0,1].
As each O_U is open, it's a union of a set I_U, of open intervals.
For each U in C, O_U = CUP I_U.
K = CUP { I_U : U in C } is an open cover of [0,1].
Since [0,1] is compact there's a finite K_f subcover of K.

Let I_0 be the interval largest interval in K_f with 0 in I0.
Set t_0 = 0.
Let I_1 be the interval in K_f with the largest upper limit that
intersects I_0. That's possible because [0,1] is connected.
Take t_1 to be any point in I_0 cap I_1 larger than t_0.

Continue this way until you get to an interval containing 1.
We see that for each j, t_(j-1) and t_j are in I_j and each I_j
is a subset of O_U for some U in C. Finally note that for each
j, f(a,t_(j-1)) and f(a,t_j) are both in some open U in C.