Topological Spaces

Definition. A topological space is a pair $(X, \tau)$ where $X$ is a set and $\tau$ is a collection of subsets of $X$ called open sets, satisfying:

1. $\emptyset \in \tau$ and $X \in \tau$.
2. The union of any sub-collection of $\tau$ is in $\tau$ (arbitrary unions of open sets are open).
3. The intersection of any finite sub-collection of $\tau$ is in $\tau$ (finite intersections of open sets are open).

The collection $\tau$ is called a topology on $X$.

Example 2.1. Let $X = \{a, b, c\}$. The collection

$\tau = \{\emptyset, \{a\}, \{a, b\}, \{a, b, c\}\}$

is a topology on $X$.

Example 2.2 (Discrete topology). For any set $X$, let $\tau_d = \mathcal{P}(X)$ (all subsets of $X$). This is the discrete topology — every subset is open.

Example 2.3 (Indiscrete topology). For any set $X$, let $\tau_i = \{\emptyset, X\}$. This is the indiscrete topology (or trivial topology).

Example 2.4 (Cofinite topology). For any infinite set $X$, let $\tau_c$ consist of $\emptyset$ and all subsets $U \subseteq X$ such that $X \setminus U$ is finite. This is the cofinite topology.

Example 2.5 (Standard topology on $\mathbb{R}$). A subset $U \subseteq \mathbb{R}$ is open if for every $x \in U$ there exists $\varepsilon > 0$ such that $(x - \varepsilon, x + \varepsilon) \subseteq U$. The collection of all such open sets is the standard topology on $\mathbb{R}$.

Definition. A basis for a topology $\tau$ on $X$ is a collection $\mathcal{B} \subseteq \tau$ such that every open set is a union of elements of $\mathcal{B}$.

Equivalently, $\mathcal{B}$ is a basis if and only if:

1. For each $x \in X$, there exists $B \in \mathcal{B}$ with $x \in B$.
2. If $B_1, B_2 \in \mathcal{B}$ and $x \in B_1 \cap B_2$, then there exists $B_3 \in \mathcal{B}$ with $x \in B_3 \subseteq B_1 \cap B_2$.

Example 2.6. The collection of all open intervals $(a, b)$ in $\mathbb{R}$ forms a basis for the standard topology.

Example 2.7. The collection of all open balls $B_r(p) = \{x \in \mathbb{R}^n : \|x - p\| < r\}$ forms a basis for the standard topology on $\mathbb{R}^n$.

Definition. Let $(X, \tau)$ be a topological space and $Y \subseteq X$. The subspace topology on $Y$ is

$\tau_Y = \{U \cap Y : U \in \tau\}.$

Proposition 2.1. If $\mathcal{B}$ is a basis for $\tau$, then $\{B \cap Y : B \in \mathcal{B}\}$ is a basis for the subspace topology on $Y$.

Example 2.8. The subspace topology on $[0, 1] \subseteq \mathbb{R}$ (with the standard topology) has $[0, 1/2)$ as an open set (since $[0, 1/2) = (-1/2, 1/2) \cap [0, 1]$).

Definition. Let $\tau_1$ and $\tau_2$ be topologies on $X$. We say $\tau_1$ is coarser (weaker) than $\tau_2$ (or $\tau_2$ is finer (stronger) than $\tau_1$) if $\tau_1 \subseteq \tau_2$.

For any set $X$: indiscrete $\subseteq$ cofinite $\subseteq$ standard (if $X = \mathbb{R}$) $\subseteq$ discrete.

These examples span the spectrum from finest (discrete, many open sets) to coarsest (indiscrete, few open sets).

• Assuming all topologies are metric. The cofinite topology on an infinite set is not Hausdorff ($T_2$) and cannot arise from a metric. Fix: Metric spaces are a proper subclass of topological spaces; many useful topologies are non-metrisable.
• Thinking arbitrary intersections of open sets are open. Only finite intersections are guaranteed. Fix: Counterexample: in $\mathbb{R}$, $\bigcap_{n=1}^\infty (-1/n, 1/n) = \{0\}$, which is not open.
• Confusing open and closed sets. A set can be both (clopen: $\emptyset$ and $X$), neither, or one without the other. Fix: Topology defines open sets; closed sets are complements. Check the definition, not intuition from $\mathbb{R}$.
• Assuming the subspace topology is intuitive. A set can be open in $Y \subseteq X$ without being open in $X$. Fix: $[0, 1/2)$ is open in $[0,1]$ with subspace topology because $[0,1/2) = (-1/2, 1/2) \cap [0,1]$.

• Data analysis (persistent homology): Topological data analysis uses simplicial complexes and persistent homology to study shape in high-dimensional data, with the Vietoris-Rips complex topology.
• Network theory: Graphs are 1-dimensional CW complexes; topological invariants like the fundamental group detect holes and cycles in communication and social networks.
• Robot motion planning: The configuration space of a robotic arm is a topological space; its connected components determine reachable configurations and its fundamental group encodes obstacles.
• Quantum computing: The topology of quantum error-correcting codes (surface codes, toric codes) determines their error thresholds; logical qubits correspond to non-contractible loops on the code manifold.
• General relativity: Spacetime is a 4-dimensional Lorentzian manifold (a topological space locally modelled on $\mathbb{R}^4$); global topology determines possible causal structures and singularities.

Problem. Let $X = \{1,2,3\}$. List all possible topologies on $X$ that are strictly finer than $\tau_1 = \{\emptyset, X\}$ and strictly coarser than $\tau_2 = \mathcal{P}(X)$. How many are there?

Solution. $\tau_1$ is the indiscrete topology (2 open sets). $\tau_2$ is the discrete topology (8 open sets). Any topology strictly between them must have at least 3 and at most 7 open sets.

We need $\tau$ such that $\tau_1 \subsetneq \tau \subsetneq \tau_2$. By checking all collections that satisfy the three axioms, we find three such topologies:

• $\tau_a = \{\emptyset, \{1\}, X\}$ (coarsest refinement)
• $\tau_b = \{\emptyset, \{1,2\}, X\}$
• $\tau_c = \{\emptyset, \{1\}, \{2\}, \{1,2\}, X\}$ (contains two singletons)

Each is strictly finer than the indiscrete topology (adds at least one non-trivial open set) and strictly coarser than the discrete topology (omits at least one singleton). There are three distinct intermediate topologies up to relabelling of points.

$\blacksquare$