Is connectedness needed in this statement?

Question

Suppose $M$ is a arcwise connected metric space, $A_1$ and $A_2$ are mutually separated in $M$. Let $f_i:A_1\cup A_2\to M$ for $i=$1,2 be two continuous functions such that $f_0|$$A_i$ ($f_0$ restricted to $A_i$) is homotopic to $f_1|A_i$ for $i=$1,2. Then $f_0$ is homotopic to $f_1$

• Here is the link about arcwise connectedness.
• Two sets A and B in a metric space are said to be mutually separated if They are disjoint and open in their union $A\cup B$
• Here is the link about homotopy

My question is

"Is arcwise connectedness really needed here to prove the above statement ?"

It seems arcwise connectedness is dispensable by below argument

Since $f_0|A_i$ homotopic to $f_1|A_i$, for $i=1,2$ There exists homotopy, $F_i:A_i×[0,1]\to M$ for $i=1,2$

Now define, $F:A_1\cup A_2×[0,1]\to M$ as $F(x,t)= \begin{cases} F_1(x,t),& \text{if } x\in A_1\\ F_2(x,t), & otherwise \end{cases} $

Here $F$ is well defined since $A_1$ and $A_2$ are disjoint. Also $F$ is a homotopy here irrespective of $M$ being arcwise connected.

Am I correct ?

Answer 1

Yes, you are correct. The assumption that $M$ is arcwise connected is completely irrelevant (as is the assumption that $A_1$ and $A_2$ are subsets of $M$, and the assumption that $M$ is a metric space). The way I would state the result with no irrelevant hypotheses would be:

Let $X$ and $M$ be topological spaces and $f_0,f_1:X\to M$ be continuous. Suppose $A_1$ and $A_2$ are disjoint open subsets of $X$ whose union is $X$, and that $f_0|_{A_i}$ is homotopic to $f_1|_{A_i}$ for $i=1,2$. Then $f_0$ is homotopic to $f_1$.