Topological Spaces

2.1 Definition

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 $X$.

2.2 Examples of Topologies

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 $\mathbb{R}$.

2.3 Basis for a Topology

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 $\mathbb{R}^n$.

2.4 Subspace Topology

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 $[0, 1/2) = (-1/2, 1/2) \cap [0, 1]$).

2.5 Comparison of Topologies

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.