I am preparing for my first course in topology. My textbook defines the components of a topological space as the equivalence classes of an equivalence relation defined by $x \sim y$ iff $x$ and $y$ are in some connected set.

I have that $\mathbb{R}$ with the standard topology or the finite complement or the countable complement topology would have exactly one component. Namely $\mathbb{R}$ itself as these are connected spaces.

I have that $\mathbb{Q}$ and $\mathbb{I}$ (irrationals) with the subspace topology have single point sets as components. Also $\mathbb{R}$ with upper/lower limit topology has all single point components.

Can someone provide some other examples of nonfinite disconnected topological spaces whose components are not trivial?

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    $\begingroup$ $X = \mathbb{R} - \{0\}$ with the subspace topology, the connected components are $(-\infty,0)$ and $(0, \infty)$ $\endgroup$ – D_S Jan 15 '17 at 20:20

Let $T$ be the group of diagonal invertible $n$ by $n$ matrices with coefficients in $\mathbb{C}$:

$$T = \{ \begin{pmatrix} x_1 \\ & \ddots \\ & & x_n \end{pmatrix} : x_i \in \mathbb{C}^{\ast} \}$$

We can identify $T$ as a subset of $\mathbb{C}^n$, giving it the induced topology.

Let $\phi: T \rightarrow \mathbb{C}^{\ast}$ be the function which sends a matrix $\textrm{Diag}(x_1, ... , x_n)$ to $x_1^4$. Let

$$S = \{ A \in T : \phi(A) = 1 \}$$

Then $S$ is a subgroup of $T$. The subgroup of $S$

$$S^0 = \{ \textrm{Diag}(1,x_2, ... , x_n) : x_i \in \mathbb{C}^{\ast} \}$$

is a connected component of $S$. The other three connected components are the nontrivial cosets of $S^0$ in $S$. They are described using the same formula as $S^0$, except the $1$ is replaced by $-1, i,$ and $-i$.


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