Definition of a Topological Space

February 17, 2026 · Luciano Muratore

Definition of a Topological Space

Topology provides an abstract framework to formalize the notion of closeness, continuity, and convergence, without relying on distances.

The central object of study is a topological space.

Definition — Topological Space

Let $X$ be a nonempty set.

A topology on $X$ is a collection $\mathcal{T}$ of subsets of $X$ such that:

1. Both the empty set and the whole space belong to $\mathcal{T}$:

   $$\varnothing \in \mathcal{T}, \qquad X \in \mathcal{T}.$$

2. $\mathcal{T}$ is closed under arbitrary unions:
   for any family $\{U_i\}_{i \in I} \subset \mathcal{T}$,

   $$\bigcup_{i \in I} U_i \in \mathcal{T}.$$

3. $\mathcal{T}$ is closed under finite intersections:
   for any $U_1, \dots, U_n \in \mathcal{T}$,

   $$\bigcap_{k=1}^n U_k \in \mathcal{T}.$$

The pair $(X, \mathcal{T})$ is called a topological space.

The elements of $\mathcal{T}$ are called open sets.

Remarks on the Definition

• No notion of distance is assumed.
• Only the behavior of unions and intersections is specified.
• The axioms are minimal and independent.
• Many analytical concepts (continuity, convergence, compactness) can be defined using only this structure.

Example 1 — Discrete Topology

Let $X$ be any set.

Define

the power set of $X$.

Verification

1. $\varnothing \in \mathcal{P}(X)$ and $X \in \mathcal{P}(X)$.
2. Arbitrary unions of subsets of $X$ are subsets of $X$.
3. Finite intersections of subsets of $X$ are subsets of $X$.

Therefore, $(X, \mathcal{T}_{\text{disc}})$ is a topological space.

Interpretation

• Every subset is open.
• Every function from a discrete space is continuous.
• This is the finest possible topology on $X$.

Example 2 — Topology Induced by a Metric

Let $(X,d)$ be a metric space.

Define $\mathcal{T}_d$ as the collection of all subsets $U \subset X$ such that:

For every $x \in U$, there exists $\varepsilon > 0$ with

$$B(x,\varepsilon) \subset U,$$

where

$$B(x,\varepsilon) = \{y \in X : d(x,y) < \varepsilon\}.$$

Verification

1. $\varnothing$ and $X$ are open by definition.
2. Arbitrary unions of open sets remain open.
3. Finite intersections of open sets remain open.

Hence, $(X, \mathcal{T}_d)$ is a topological space.

Interpretation

• This topology encodes the notion of proximity given by the metric.
• All metric notions of convergence and continuity are preserved.
• Many different metrics can induce the same topology.

Final Remarks

• A topology abstracts the structure behind open sets.
• Metric spaces are special cases of topological spaces.
• Topology allows us to study continuity and convergence without numerical distance.

This definition is the foundation upon which modern analysis and geometry are built.