General Topology: The Theory of Convergence and Continuity

General Topology (or Point-Set Topology) is the study of the structure of space. It provides the most general framework for discussing limits, continuity, and connectedness. While Analysis often relies on the notion of distance provided by a metric, Topology abstracts these concepts to collections of sets, allowing us to study spaces where “distance” may be undefined or irrelevant.

1. The Definition of a Topological Space $(X,\tau )$

A Topological Space is an ordered pair $(X,\tau )$, where $X$ is a set and $\tau$ is a collection of subsets of $X$ (called open sets) satisfying the following three axioms:

1. Entirety: The empty set $\varnothing$ and the set $X$ itself are in $\tau$.
2. Arbitrary Unions: The union of any sub-collection of sets in $\tau$ is also in $\tau$: ${\bigcup }_{i\in I}{U}_i\in \tau \text{for }\{U_i{\}}_{i\in I}\subseteq \tau$
3. Finite Intersections: The intersection of any finite sub-collection of sets in $\tau$ is also in $\tau$: ${\bigcap }_{i=1}^n{U}_i\in \tau \text{for }\{U_1,\dots ,U_n\}\subseteq \tau$

The collection $\tau$ is called the topology on $X$. A subset $C\subseteq X$ is called closed if its complement $X\setminus C$ is open.

2. Basis of a Topology

Often, it is cumbersome to list every open set. Instead, we define a topology using a Basis. A collection $\mathcal{B}$ of subsets of $X$ is a basis for a topology if:

1. For every $x\in X$, there is at least one basis element $B\in \mathcal{B}$ such that $x\in B$.
2. If $x\in B_1\cap B_2$ for $B_1,B_2\in \mathcal{B}$, there exists a $B_3\in \mathcal{B}$ such that $x\in B_3\subseteq B_1\cap B_2$.

The topology generated by $\mathcal{B}$ consists of all sets $U$ that can be written as a union of elements of $\mathcal{B}$.

3. Continuity: The Topological Perspective

In calculus, we define continuity using $ϵ-\delta$ limits. In topology, we generalize this: A function $f:X\to Y$ is continuous if and only if for every open set $V\subseteq Y$, the preimage $f^{-1}(V)$ is open in $X$.

$f\text{ is continuous}⟺\forall V\in {\tau }_Y,f^{-1}(V)\in {\tau }_X$

This definition is powerful because it requires no arithmetic—only the structure of open sets.

4. Homeomorphisms: Topological Equivalence

A Homeomorphism is a bijection $f:X\to Y$ such that both $f$ and $f^{-1}$ are continuous. If such a mapping exists, $X$ and $Y$ are said to be homeomorphic.

Technically, a homeomorphism is an isomorphism in the category of topological spaces. It preserves all “topological properties” (properties that depend only on the topology, like compactness or connectedness). This is why a coffee cup and a donut (a torus) are topologically equivalent: one can be continuously deformed into the other without tearing or gluing.

5. Separation Axioms

Separation axioms describe how well we can distinguish distinct points or sets using open sets.

• $T_1$ Space: For any two distinct points $x,y$, there exist open sets $U,V$ such that $x\in U,y\notin U$ and $y\in V,x\notin V$.
• $T_2$ (Hausdorff) Space: For any two distinct points $x,y$, there exist disjoint open sets $U$ and $V$ such that $x\in U$ and $y\in V$. Most “natural” spaces in analysis (like ${\mathbb{R}}^n$) are Hausdorff. In Hausdorff spaces, limits of sequences are unique.
• $T_3$ (Regular): $X$ is $T_1$ and for any closed set $A$ and point $x\notin A$, there exist disjoint open sets $U$ and $V$ such that $A\subseteq U$ and $x\in V$.
• $T_4$ (Normal): $X$ is $T_1$ and for any two disjoint closed sets $A,B$, there exist disjoint open sets $U,V$ such that $A\subseteq U$ and $B\subseteq V$.

6. Compactness

Compactness generalizes the notion of being “finite” or “bounded” to abstract spaces. A space $X$ is compact if every open cover of $X$ has a finite subcover.

Formally: if $X\subseteq {\bigcup }_{\alpha \in A}{U}_{\alpha }$ where $U_{\alpha }$ are open, then there exists a finite subset $\{U_{{\alpha }_1},\dots ,U_{{\alpha }_n}\}$ such that $X\subseteq {\bigcup }_{i=1}^n{U}_{{\alpha }_i}$.

Heine-Borel Theorem: A subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded.

7. Connectedness

A space $X$ is disconnected if there exist two disjoint, non-empty open sets $U,V$ such that $X=U\cup V$. If no such separation exists, $X$ is connected.

A stronger notion is path-connectedness: $X$ is path-connected if for any two points $x,y\in X$, there exists a continuous map $\gamma :[0,1]\to X$ such that $\gamma (0)=x$ and $\gamma (1)=y$. Every path-connected space is connected, but the converse is not always true (e.g., the Topologist’s Sine Curve).

8. Metric Spaces and Metrizability

A metric $d:X\times X\to \mathbb{R}$ induces a topology by defining a basis of open balls: $B(x,ϵ)=\{y\in X:d(x,y)<ϵ\}$

A topological space is metrizable if there exists a metric that induces its topology. Urysohn’s Metrizability Theorem states that every second-countable regular ($T_3$) space is metrizable.

9. Product and Quotient Topologies

• Product Topology: Given spaces $X$ and $Y$, the product topology on $X\times Y$ is generated by the basis $\mathcal{B}=\{U\times V:U\in {\tau }_X,V\in {\tau }_Y\}$.
• Quotient Topology: Given a space $X$ and an equivalence relation $\sim$, the quotient map $q:X\to X/\sim$ induces a topology on the set of equivalence classes $X/\sim$ where $U$ is open in $X/\sim$ iff $q^{-1}(U)$ is open in $X$. This is used to construct objects like the Möbius strip or the Klein bottle.

Python Visualization: Overlapping Intervals as a Basis

The following script demonstrates how a collection of open intervals can “cover” a space and act as a basis for the standard topology on $\mathbb{R}$.

import matplotlib.pyplot as plt
import numpy as np

def visualize_basis(n_intervals=15):
    fig, ax = plt.subplots(figsize=(10, 3))
    
    # Generate random overlapping intervals (a, b) in [0, 1]
    for i in range(n_intervals):
        start = np.random.uniform(0, 0.7)
        width = np.random.uniform(0.1, 0.3)
        end = start + width
        
        # Represent open intervals with transparency and open markers
        ax.hlines(y=0.1 + i*0.1, xmin=start, xmax=end, 
                 colors='C0', linewidth=3, alpha=0.6)
        ax.plot(start, 0.1 + i*0.1, 'bo', markersize=6, fillstyle='none')
        ax.plot(end, 0.1 + i*0.1, 'bo', markersize=6, fillstyle='none')

    ax.set_ylim(0, n_intervals * 0.1 + 0.2)
    ax.set_xlim(0, 1)
    ax.set_yticks([])
    ax.set_title("Generating a Topology: Set of Overlapping Open Intervals (Basis Elements)")
    ax.set_xlabel("Real Line Segment [0, 1]")
    plt.tight_layout()
    plt.show()

# Run the visualization
visualize_basis(15)