Topology: Homeomorphism between finite complement topology in $\mathbb{R}$ and one of its subspaces My class notes say that because $U=\mathbb{R}\backslash\{x_1,x_2,..,x_n\}$ has the same cardinality than $\mathbb{R}$, there exists a homeomorphism between:
$(U,T_{cof})$ and $(\mathbb{R},T_{cof})$, where $T_{cof}$ is the finite complement topology.
I initially thought that having the same cardinality is a necessary condition but is not sufficient to have an homeomorphism.  Also, I can't manage to find a homeomorphism between these two.

The following is easy to prove:
Proposition 1: Let $X$ be a set endowed with the cofinite topology. The subspace topology on any $Y \subset X$ is identical to the cofinite topology on $Y$.
Proof: Exercise.
Proposition 2: Let $X$ and $Y$ be two cofinite topological spaces. Then any injective mapping $f: X \to Y$ is continuous.
Proof
The proof will be complete if we can show that
$\tag 1 \text{For any } A \subset X, \; f(\overline{A})\subseteq \overline{f(A)}$
(click here).
There are two cases:
Case 1: If $A$ is infinite, $\overline{A} = X$. Since $f$ is injective, $f(A)$ is infinite and $\overline{f(A)} = Y$.
Since $f(X) \subseteq Y$, $\text{(1)}$ must be true.
Case 2: If $A$ is finite, $\overline{A} = A$ and the lhs of $\text{(1)}$ is $f(A)$. The image $f(A)$ is also finite and the rhs of $\text{(1)}$ is also $f(A)$, and so the inclusion relation in $\text{(1)}$ must again be true.$\quad \blacksquare$
Using the above we can now state
Proposition 3: Let $f$ be an injective mapping from a set $X$ with the cofinite topology to a set $Y$ with the cofinite topology. Then $f$ is a homeomorphism between $X$ and its image $f(X)$.

Let $U=\mathbb{R}\backslash\{x_1,x_2,..,x_n\}$ for distinct numbers $x_i$ and assume that
$\quad x_n = \text{max(}x_1,x_2,..,x_n\text{)}$
Extend the finite sequence by defining $x_{n+k} = x_n + k$ for $k \ge 1$.
We can define an injection on the set of $x_i$ via
$\quad x_i \to x_{i+n}$
This injection can be easily extended to define a bijective correspondence between $\mathbb R$ and $U$.
