0
$\begingroup$

I am preparing for Master's admission interview. I found some previous year interview questions.

  1. Is there a continuous map from $[0,1]$ to $(0,1)$? Why/Why-not?

My attempt: $f(x) = \frac{1}{3}$. So, answer is yes.

  1. Is there a continuous map from $(0,1)$ to $[0,1]$?

My attempt: $f(x)= 0$ if $x\in (0, 1/4]$, $12x-3$ when $x\in [1/4, 1/3]$, and $1$ when $x\in [1/3,1)$.

  1. Construct a bijective map from $(0,1)$ to $[0,1]$.

My attempt: Let $A=\{\frac{1}{2},\frac{1}{3},...\}$,$B=\{1,\frac{1}{2},\frac{1}{3},...\}$. Define $f:A\rightarrow B$ such that $f(\frac{1}{n})=\frac{1}{n-1}$. Now define a function $g:(0,1) \rightarrow [0,1]$ such that $g(x)=x$ if $x$ is not in $A$ , otherwise $g(x)=f(x)$.

Then $g$ is a the bijection from $(0,1)$ to $(0,1]$.

Am I right?

$\endgroup$
3
  • $\begingroup$ For 1. and 2. it seems right as all constant maps are continuous. And for 3. that seems right, too. As it should be surjective and injective, then you can apply it in a similar way to 0 $\endgroup$
    – linkja
    Dec 14, 2022 at 19:34
  • $\begingroup$ @linkja Thanks. For 1. I am not happy with my solution. Can you or anyone please provide a good solution for me? $\endgroup$
    – user1116521
    Dec 14, 2022 at 19:45
  • $\begingroup$ For 3 are you looking for $(0,1)$ to $[0,1]$ or $(0,1)$ to $(0,1]$? Your answer would be correct if it goes to $(0,1].$ $\endgroup$
    – blakedylanmusic
    Dec 15, 2022 at 16:38

1 Answer 1

Reset to default
0
$\begingroup$

For 1. and 2.:

Let $X,Y$ be topological spaces and $f:X\to Y$ a constant map, $f(x) = y_0$, then $f$ is continuous, as for any $U\subset Y $ open either contains $y_0$, so $f^{-1}(U) = X$, which is open, or $U$ doesn't contain $y_0$ then $f^{-1}(U)=\emptyset$, which is also open. Therefor $f$ is continuous.
Now take you can choose $X,Y$ and $y_0$ accordingly for 1. and 2..

$\endgroup$
5
  • $\begingroup$ If this person is doing a masters admissions interview they may not have learned about topological spaces yet. $\endgroup$
    – blakedylanmusic
    Dec 15, 2022 at 16:34
  • $\begingroup$ I think in this case it is rather irrelevant if you know what a topological space is, you just need to know that continuity of a map can also characterized by the preimages of open sets being open. And $X,Y$ have to be something $\endgroup$
    – linkja
    Dec 15, 2022 at 16:44
  • $\begingroup$ Yes, I agree it’s relevant to the question because the person was checking if constant maps are continuous. But it may not be too helpful if the person hasn’t been exposed to topological spaces yet. $\endgroup$
    – blakedylanmusic
    Dec 15, 2022 at 16:46
  • $\begingroup$ How would you suggest to formulate it? $\endgroup$
    – linkja
    Dec 15, 2022 at 16:47
  • $\begingroup$ I would use the epsilon-delta definition with subsets of $\mathbb{R}$ to explain this. $\endgroup$
    – blakedylanmusic
    Dec 15, 2022 at 16:49

You must log in to answer this question.