Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true?

Question

Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true?

$a.$ If $X$ has at least $n$ distinct points, then the dimension of $C(X)$, the space of continuous real valued functions defined on $X$, is at least $n$.

$b.$ If $A$ and $B$ are disjoint, non-empty and closed sets in $X$, there exists $f \in C(X)$ such that $f(x) = −3$ for all $x ∈ A$ and $f(x) = 4$ for all $x ∈ B.$

$c$. If $A ⊂ X$ is a closed and non-empty subset and if $g : A → R$ is a continuous function, then there exists $f ∈ C(X)$ such that $f(x) = g(x)$ for all $x ∈ A.$

I can conclude that part C is true....but not conclude about parts a and b.. Help needed!

Answer 1

For (a) order the n points as $\{x_1, x_2, ..., x_n\}$. Define $f_i$ by "$f_i(x_n)$ equals 1 if n= i, 0 otherwise". And extend to all other points by "continuity"- that is, so that the function is continuous. Show that these n functions are independent.

Answer 2

Indeed c) follows from the Tietze extension theorem plus the fact that compact Hausdorff spaces are normal.

Then a) and b) also follow from the same theorem: apply the theorem to the closed set $A \cup B$ and a function that is $-3$ on $A$ and $4$ on $B$, which is continuous on $A \cup B$ (you have constant functions on two components).

And a) follows from the argument by user247327: $A = \{x_1,\ldots,x_n\}$ is closed, and we have $n$ functions $f_i$ defined on $A$ ($f_i(x_j) = 1$ iff $i = j$, otherwise $0$), that we extend by Tietze, and are linearly independent in $C(X)$.