I am having difficulty dealing with the problem below. The problem is from Munkrees Topology text. Can someone please help me solving the problem? $\def\R{{\mathbb R}} \def\N{{\mathbb N}}$
Find a compactification $K(X)$ for the indicated space $X$ as required. Let $E(X)$ denote the remainder, so $E(X) = K(X) - X$.
(b) $X=\N$ and $E(X)$ consists of exactly three isolated points.
$\textbf{Solution:}$ Let us use the three disjoint infinite subsets $N_1,N_2,N_3$ of $\Bbb N$ and add $p_1,p_2,p_3$ as new points to $\Bbb N$ and use cofinite subsets of $N_i$ plus $\{p_i\}$ as a local base at $p_i$, generalising the one-point compactification idea. Disjointness is needed for Hausdorffness of the resulting space.
(c) $X=\N$ and $E(X)$ is homeomorphic to $[0,1]$.
$\textbf{Solution:}$ Suppose $X=([0,1] \times \{0\}) \cup \{(q_n, \frac{1}{n+1}): n \in \omega\}$ where $q_n, n \in \omega$ is an enumeration of the rationals in $[0,1]$, as a subspace of the plane. The scattered points above the interval are discrete, so homeomorphic to $\Bbb N$, and $X$ is compact.
For c) try $X=([0,1] \times \{0\}) \cup \{(q_n, \frac{1}{n+1}): n \in \omega\}$ where $q_n, n \in \omega$ is an enumeration of the rationals in $[0,1]$, as a subspace of the plane. The scattered points above the interval are discrete, so homeomorphic to $\Bbb N$, and $X$ is compact.
For b) use the three disjoint infinite subsets $N_1,N_2,N_3$ of $\Bbb N$ and add $p_1,p_2,p_3$ as new points to $\Bbb N$ and use cofinite subsets of $N_i$ plus $\{p_i\}$ as a local base at $p_i$, generalising the one-point compactification idea. Disjointness is needed for Hausdorffness of the resulting space.
For a) I have some ideas, but please think for yourself first..
Fun fact: for $\Bbb R$ the remainder can only be finite if it has 1 or 2 points.