Simple proof there is no continuous bijection from $mathbb{R}^n$ onto $[0,1]^m$

I improved the my previous idea, but it's too long to write a comment, so I decided to post it here. If anything goes wrong, I will delete my answer or edit it. Actually, I'm suspicious that I missed something because this question seems to be more complicated....

Assume $(Bbb{R}^n,d_1)$ and $([0,1]^m,d_2)$ be the two metric spaces.

Suppose $exists$ such $f$ and Let $x_0=(a_1,a_2,...,a_n)inBbb{R}^n$ s.t. $f(x_0)=(1,1,...,1)$. Then, since both spaces are metric spaces, we see that $d_1(x_0,y)<epsilonimplies d_2(f(x_0),f(y))<delta$ by continuity. So, in general, we can take two sets $U,V$ containing $x_0$, where $U=prod_{i=1}^{n-1}(a_i-epsilon,a_i+epsilon)times(a_n-epsilon,a_n]$ and $V=prod_{i=1}^{n-1}(a_i-epsilon,a_i+epsilon)times[a_n,a_n+epsilon)$. Their images $f(U),f(V)subset(1-delta,1]^m$.

Case 1: If $f(U)cap f(V)=varnothing$

Suppose it's possible, then each point in $f(Ucap V)=f(prod_{i=1}^{n-1}(a_i-epsilon,a_i+epsilon)times a_n)$ would have two distinct images when it's in $U$ and in $V$, which contradicts the assumption that $f$ is bijective.

Case 2: If $f(U)cap f(V)cap ((1-delta,1]^msetminus f(Ucap V))neqvarnothing$

Then there must exist repetitions that undermines the assumption about bijectivity.

Case 3: If $f(U)cap f(V)=f(Ucap V)$

Then we let $$U=prod_{j=1}^{n-2}(a_j-epsilon,a_j+epsilon)times (a_{n-1}-epsilon,a_{n-1}]times a_n$$ and $$V=prod_{j=1}^{n-2}(a_j-epsilon,a_j+epsilon)times [a_{n-1},a_{n-1}+epsilon)times a_n.$$ By doing this, we restricts the image to an $(m-1)$ dimensional cube containing $(1,1,...,1)$, so we face exactly the same three situations again, and a similar argument push this to the $(m-2)$ dimension, where $U$ becomes $$prod_{j=1}^{n-3}(a_j-epsilon,a_j+epsilon)times (a_{n-2}-epsilon,a_{n-2}]timesprod_{j=n-1}^na_j$$ and $V$ becomes $$prod_{j=1}^{n-3}(a_j-epsilon,a_j+epsilon)times [a_{n-2},a_{n-2}+epsilon)timesprod_{j=n-1}^na_j.$$

We repeat that process with new $U,V$ on and on until the images get to dimension one, which reduces to the situation of $Bbb{R}^nto lapprox[0,1]$ (restricting the range to a line segment). Their images, (when restricted to $l$), should be $(a,1]$ and $(b,1]$ (the notation is a little bit weird... but should be half open segments) respectively, where $1-delta<a,b<1$. then the intersection must be nontrivial and contain extra points.

I missed the space-filling curve, as Cronus pointed out in his comment. The main obstruction of this method is that I've not found any way to argue the fractal curves. I'm still thinking, but maybe someone could help me out.

I thought about the reason why we need to reduce the dimensions: if $m>1$, then we always get a plane or something that $U$ and $V$ may not be able to intersect as desired because the image has various directions to go, but in $m=1$, there is only one direction, so the image of $U$ and $V$ will certainly intersect each other that contains extra points.

I know that this part is hard to understand. It's also hard to describe clearly, so I think we may need further discussion on that......

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I'm not really sure if this is valid, but it'll never get verified and edited unless I post it. If someone found flawness in this answer, please kindly point out and I'll reconsider or delete my post.

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Edit:

Actually, if both spaces were equipped with the order topology built on the usual dictionary order, then this would be easy.

Assume $f$ is conts bijective, then $exists x,yinBbb{R}^n$ s.t. $f(x)=(0,...,0)$, $f(y)=(1,...,1)$. Because $Bbb{R}^n$ is connected and $[0,1]^m$ is assumed to be an ordered set in the order topology, we can apply the (generalized version) intermediate value theorem: Since in the order topology, the interval $(f(x),f(y))$ must go through all points in the cube, which corresponds to the image of the interval $(x,y)$. Now, the point in $Bbb{R}^nsetminus (x,y)$ needs to have image but it will undermines the bijectivity, which gives us a contradiction.