The importance of the condition $\mathcal{B}$ refines $\mathcal{A}$ in finding a function $\phi$ making the diagram commutes.

Question

If I know the following definitions and how to prove(a) and (b) in the following problem:

DEFINITION.

If $\mathcal{B}$ is a disjoint cover of $X,$ then we can define a function $$f_{\mathcal{B}}: X \rightarrow \mathcal{B},$$given by the formula $$f_{\mathcal{B}}(x) = \textbf{the unique $B \in \mathcal{B}$ such that $x \in B$}.$$And to get some topology involved, we give $\mathcal{B}$ the discrete topology.

DEFINITION.

Suppose $\mathcal{A}$ and $\mathcal{B}$ are two covers of $X.$ We say that $\mathcal{B}$ refines $\mathcal{A}$ if each member of $\mathcal{B}$ is contained in some member of $\mathcal{A}.$ We say that $\mathcal{B}$ strictly refines $\mathcal{A}$ if each member of $\mathcal{B}$ is a proper subset of some member of $\mathcal{A}.$ if $\mathcal{B}$ strictly refines $\mathcal{A},$ we write $\mathcal{A} < \mathcal{B}.$

problem:

Let $X$ be a space and let $\mathcal{B}$ be a disjoint cover of $X.$

$(a)$ Show that $f_{\mathcal{B}}$ is continuous iff $\mathcal{B}$ is a clopen cover of $X.$

$(b)$ Suppose $\mathcal{A}$ and $\mathcal{B}$ are both disjoint clopen covers of $X,$ so that we have two continuous functions $f_{\mathcal{A}}: X \rightarrow \mathcal{A}$ and $f_{\mathcal{B}}: X \rightarrow \mathcal{B}.$ Show that there can be at most one function $\phi : \mathcal{B} \rightarrow \mathcal{A}$ making the diagram below commute(i.e., such that $\phi \circ f_{\mathcal{B}} = f_{\mathcal{A}}$)

Now, I want to prove that:

$(c)$ Show that the dotted arrow can be filled with a function $\phi$ making the diagram commute if and only if $\mathcal{B}$ refines $\mathcal{A}.$

Note that I know also the proof of this problem:

Let $X$ be a compact metric space that is totally disconnected, and let $\epsilon > 0.$

(a) Show that $X$ has a finite cover $\mathcal{A}$ clopen sets with diameter at most $\epsilon.$

(b) Show that there is a clopen cover $\mathcal{B}$ such that $\mathcal{B}$ refines $\mathcal{A}$ and distinct numbers of $\mathcal{B}$ are disjoint.

How can I prove the required above (I mean $(c)$ in the first problem)?

Answer 1

HINT: Think about what it means for the diagram to commute: for each $x\in X$ we must have $f_{\mathscr{A}}(x)=\varphi(f_{\mathscr{B}}(x))$. Now what is $f_{\mathscr{A}}(x)$? It’s the unique member of $\mathscr{A}$ that contains the point $x$. Similarly, $f_{\mathscr{B}}(x)$ is the unique member of $\mathscr{B}$ that contains $x$. We need a map $\varphi:\mathscr{B}\to\mathscr{A}$ with the following property:

• whenever $x\in B\in\mathscr{B}$, then $x\in\varphi(B)\in\mathscr{A}$.

In words, if $B\in\mathscr{B}$, every point of $B$ must also belong to $\varphi(B)$. What does that say about the relationship between the two sets $B$ and $\varphi(B)$?

There’s a bit more work to be done, but that’s really the key to proving both directions of (c).