Problem: The interval $[a,b]$ $\subset$ $\mathbb{R}$ and interval $[0,1]$ $\subset$ $\mathbb{R}$ are homeomorphic in their standard topologies. Definition of Homeomorphism: We call the underlying bijection of a topological equivalence $h : X \to Y$ a homeomorphism. We say X and Y are homeomorphic. This is how I showed it. Let us define a function $f:[a,b]$$\to$$[0,1]$ […]

- Tags \infty)$ ?) in standard topology, $f(x_1)$ = $f(x_2)$, $f(x_n) \to f(x)$. Therefore, $f(x_n) = \frac{x_n-a}{b-a}$ $\to$ $\frac{x-a}{b-a}$ [Since, $f$ is homeomorphism and $[a, $f$ is injective or one-to-one. Let, $x_1$ = $x_2$. Thus, $x_n \to a$]. Therefore, $x_n$ be a sequence in $[a, 1]$ $\subset$ $\mathbb{R}$ are homeomorphic in their standard topologies. Definition of Homeomorphism: We call the underlying bijection of a, 1]$ are homeomorphic. (Proved) Please note my prof hasn't done compact sets so far so it will be a problem if I use that concept while provi, 1]$ be any element and $f(x) = y$, 1]$ by $f(x)$ = $\frac{x-a}{b-a}$, 1]$ has a pre-image in $[a, 1]$ is $T_2$ in usual topology, a, a) \cup (b, and also $[0, B, b]$ $\subset$ $\mathbb{R}$ and interval $[0, b]$ and $[0, b]$ as $a(1-y)+by$ is a pre-image of $y$. Now, b]$ is a compact set (can I say it has standard (order) topology b/c its sub-basis consists of positive and negative open rays or $(-\infty, b]$ is a convex set, b]$. As, b]$$\to$$[0, clearly $f$ is surjective. Hence, helps me on notation and also explains me to show its homeomorphic without using compact set at the end. Appreciate your continuous support., it is bijective. We now prove that $f$ is continuous. Let, it is continuous. Since, it would be great if someone verifies my proof, Problem: The interval $[a, so each element in $[0, such that $x_n \to x$. Then, then => $\frac{x-a}{b-a}$ = $y$ => $x = (b-a)y + (x-a)$ = $a(1-y)+by$ $\in$ $[a, then $\frac{x_1-a}{b-a}$ = $\frac{x_2-a}{b-a}$, then we know that clearly f(x) is well defined. Let, therefore from that we get that $x_1-a$ = $x_2-a$ (when b $\neq$ a). Therefore, we can see that, we know that any continuous bijection from a compact space to a topological space is homeomorphism. Hence, we want to prove that $f(x_n) \to f(x)$. Now, y) \in [0