|
|
$\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\U}{\mathcal{U}}$- Any set $X$ endowed with the trivial topology, where the only open sets are $\emptyset$ and $X$.
- Any set $X$ endowed with the discrete topology, where all sets are open.
- The Sierpiński space $\{0,1\}$ with topology $\{\emptyset, \{1\}, \{0,1\}\}$.
- Any non-empty set $X$ containing a point $p$ endowed with the particular point topology, where the only open sets are the empty set and those sets that contain $p$.
- $[0,1]$ with the standard topology.
- $(0,1)$ with the standard topology.
- $[0,1)$ with the standard topology (is this homeomorphic to $\R$?).
- $[0,1]$ with the either-or topology, where a set is open if it either does not contain $\{0.5\}$ or it does contain $(0,1)$.
- The circle.
- $\N$ with the cofinite topology, where a set is open if it is empty or if its complement is finite.
- $\R$ with the cocountable topology, where a set is open if it is empty or if its complement is countable.
- The topologist's sine curve, $\{(x, \sin (1/x)) \mid x \in (0,1]\} \cup \{(0,0)\}$, with the subspace topology inherited from the Euclidean plane.
- The Sorgenfrey line, which is $\R$ with the topology generated by the sets $[a,b)$.
- The Sorgenfrey plane, which is the product of the Sorgenfrey line with itself.
- The long ray, which is the cartesian product of the first uncountable ordinal $\omega_1$ with $[0,1)$. This set has the lexicographic order where $(a,b) < (c,d)$ if $a < c$ or if $a = c$ and $b < d$. It is endowed with the order topology, which is generated by the sets $\{x \mid a < x \}$ and $\{x \mid x < b\}$.
- The long line, which is constructed from the long ray in the same way that $(-1,1)$ is constructed from $[0,1)$.
- The Tychonoff plank, which is the product of the ordinal spaces $[0,\omega]$ and $[0, \omega_1]$.
- The infinite broom, which is the subspace of the Euclidean plane consisting of all closed line segments joining the origin to $(1, 1/n)$ for all positive integers $n$, together with the interval $(0.5, 1]$ on the $x$-axis.
- The integer broom, which is the subset of the Euclidean plane given by the polar coordinates $\{r = n \mid n \in \N_0\} \times \{\theta = 1/k \mid k \in \N, k \geq 1\} =: U \times V$. It has the product topology generated by the standard topology on $V$ and the topology on $U$ generated by the sets $\{n \mid a < n\}$.
- The Cantor set.
- The Hilbert cube, which is the topological product $\prod_{i=1}^n [0,1/n]$.
|
|
