Examples

• \({\mathbf{Q}}\subseteq {\mathbf{R}}\) is neither open nor closed:
  • No point is interior, since every neighborhood of \(q\in {\mathbf{Q}}\) intersects some \(p\in {\mathbf{R}}\setminus{\mathbf{Q}}\). No point in the complement is interior for a similar reason.
  • It has no isolated points, and every \(r\in {\mathbf{R}}\) is a boundary and accumulation point.
• \({\mathbf{Z}}\subseteq {\mathbf{R}}\) is closed and not open.
  • Just write \({\mathbf{R}}\setminus{\mathbf{Z}}= \cup_{n\in {\mathbf{Z}}} (n, n+1)\), a countable union of open sets. There are no interior points: every neighborhood of \(n\in {\mathbf{Z}}\) intersects \({\mathbf{R}}\setminus{\mathbf{Z}}\).
  • Every point is a boundary and accumulation point.
• Points are closed in \({\mathbf{R}}\): \({\mathbf{R}}\setminus\left\{{ p }\right\} = (-\infty, p) \cup(p, \infty)\).
  • An infinite intersection of open sets need not be open: \(\cap_{n\in {\mathbb{N}}} (p-1/n, p+1/n) = \left\{{ p }\right\}\) which is closed.
• Intervals \((a, b)\) are open in \({\mathbf{R}}^1\) but not in \({\mathbf{R}}^d\) for \(d\geq 2\).
• \(\left\{{1/n}\right\}\) has only isolated boundary points, and no interior points. The point \(0\) is an accumulation point.
• The Cantor set has no interior points and no isolated points. Every point is a boundary point and an accumulation point.

The following are some standard “nice” spaces: \begin{align*} S^n, {\mathbb{D}}^n, T^n, {\mathbf{RP}}^n, {\mathbf{CP}}^n, \mathbb{M}, \mathbb{K}, \Sigma_{g}, {\mathbf{RP}}^\infty, {\mathbf{CP}}^\infty .\end{align*}

The following are useful spaces to keep in mind to furnish counterexamples:

• Finite discrete sets with the discrete topology.
• Subspaces of \({\mathbf{R}}\): \((a, b), (a, b], (a, \infty)\), etc.
  • Sets given by real sequences, such as \(\left\{{0}\right\} \cup\left\{{{1 \over n}{~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 1}}\right\}\)
• \({\mathbf{Q}}\)
• The topologist’s sine curve
• One-point compactifications
• \({\mathbf{R}}^\omega\) for \(\omega\) the least uncountable ordinal (?)
• The Hawaiian earring
• The Cantor set

Examples of some more exotic spaces that show up less frequently:

• \({\operatorname{HP}}^n\), quaternionic projective space
• The Dunce Cap
• The Alexander Horned sphere

The following spaces are non-Hausdorff:

• The cofinite topology on any infinite set.
• \({\mathbf{R}}/{\mathbf{Q}}\)
• The line with two origins.
• Any variety \(V(J) \subseteq {\mathbf{A}}^n_{/k}\) for \(k\) a field and \(J{~\trianglelefteq~}k[x_1, \cdots, x_{n}]\).

The following are some examples of ways to construct specific spaces for examples or counterexamples:

• Knot complements in \(S^3\)

• Covering spaces (hyperbolic geometry)

• Lens spaces

• Matrix groups

• Prism spaces

• Pair of pants

• Seifert surfaces

• Surgery

• Simplicial Complexes

  • Nice minimal example:

The following are some nice examples of topologies to put on familiar spaces to produce counterexamples:

• Discrete
• Cofinite
• Discrete and Indiscrete
• Uniform

The cofinite topology on any space \(X\) is always

• Non-Hausdorff
• Compact

Some facts about the discrete topology:

• Definition: every subset is open.
• Always Hausdorff
• Compact iff finite
• Totally disconnected
• If \(X\) is discrete, every map \(f:X\to Y\) for any \(Y\) is continuous (obvious!)

Some facts about the indiscrete topology:

• Definition: the only open sets are \(\emptyset, X\)
• Never Hausdorff
• If \(Y\) is indiscrete, every map \(f:X\to Y\) is continuous (obvious!)
• Always compact