Topology

Finite Topology Library

A Python library for defining, analyzing, and exploring finite topologies. This toolset is designed for educational purposes, to assist in research, and to make it easy to experiment with topological concepts computationally, using finite sets.

Features

Construct Common Topologies:

• Discrete Topology: Every subset is open.
• Indiscrete (Trivial) Topology: Only the empty set and the entire set are open.
• Sierpiński Topology: A classic topology on a two-element set.
• Particular Point and Excluded Point Topologies: Control which points are required or excluded in open sets.
• Divisibility Topology: Topology based on divisibility relations among integers.
• Alexandrov Topology: Topologies that are closed under arbitrary intersections of open sets, useful in domain theory.

Analyze Functions Between Spaces:

• Continuity: Check whether a function between two topological spaces is continuous.
• Homeomorphism: Analyze whether two spaces are homeomorphic (topologically equivalent) via bijections.
• Preimage and Image: Compute preimages of sets under continuous mappings.

Topology Properties:

• Compactness: Determine if a topology is compact, a trivial property for finite topologies.
• Connectedness: Check if the space can be split into two disjoint open sets.
• Hausdorff Conditions (T2): Verify if any two distinct points have disjoint open neighborhoods.
• Separation Axioms (T0, T1): Study basic separation properties in finite spaces.

Installation

To install the library, run the following command:

pip install finite-topology

Alternatively, you can clone the repository and install dependencies:

git clone https://github.com/nand0san/Topology.git
cd finite-topology
pip install -r requirements.txt

Usage

Example 1: Create a Discrete Topology and Analyze a Continuous Function

from finite_topology.known_topologies import create_discrete_topology
from finite_topology.functions import Function

# Define two sets
space = {1, 2, 3}
target_space = {'a', 'b', 'c'}

# Create discrete topologies on both spaces
source_topology = create_discrete_topology(space)
target_topology = create_discrete_topology(target_space)

# Define a mapping for a continuous function
mapping = {1: 'a', 2: 'b', 3: 'c'}

# Create a function and verify continuity
f = Function(source_topology, target_topology, mapping)
print(f"Is the function continuous? {f.is_continuous()}")

Expected Output:

Is the function continuous? True

Example 2: Study Dense Sets in Finite Topologies

In finite topological spaces, dense sets exhibit interesting behavior. Consider the triviality that the entire space is dense in many cases.

from finite_topology.topology import Topology

# Define a collection of subsets that form a topology
space = {1, 2, 3}
subsets = [{1}, {1, 2}, {1, 2, 3}]
topology = Topology(collection_of_subsets=subsets, generate=False)

# Find the dense subsets
dense_set = topology.find_dense_subset()
print(f"Dense subset: {dense_set}")

Explanation: In finite topologies, a set is dense if its closure equals the entire space. The library can compute and return the smallest dense subset or indicate if the entire space is the only dense subset (trivial case).

Expected Output:

Dense subset: {1}

Example 3: Topologies with Specific Points (Particular Point Topology)

from finite_topology.known_topologies import create_particular_point_topology

space = {1, 2, 3}
particular_point = 2

# Create a particular point topology
topology = create_particular_point_topology(space, particular_point)

print(f"Topology with particular point {particular_point}:")
print(topology)

Expected Output:

Topology with particular point 2:
Topology(
  Space: [1, 2, 3],
  Collection of Subsets:
    [1, 2, 3]
    [2]
    [1, 2]
    [2, 3]
  Properties: T0, T1, Connected, Compact, Separable
)

In this example, the topology is defined such that every non-empty open set must contain the particular point.

Example 4: Checking Hausdorff Conditions (T2)

from finite_topology.known_topologies import create_indiscrete_topology

# Create an indiscrete topology
space = {1, 2, 3}
topology = create_indiscrete_topology(space)

# Check if the topology is Hausdorff
print(f"Is the topology Hausdorff? {topology.is_hausdorff()}")

Expected Output:

Is the topology Hausdorff? False

Note: In finite spaces, the only Hausdorff topology is the discrete topology, as demonstrated in this Math StackExchange discussion.

Example 5: Generating a Topology from a Subbase

Use the generate=True parameter to create a topology from a collection of subsets acting as a subbase.

from finite_topology.topology import Topology

# Define a collection of subsets that act as a subbase
subbase = [{1}, {2, 3}]

# Create the topology generated from the subbase
topology = Topology(collection_of_subsets=subbase, generate=True)

print("Topology generated from subbase:")
print(topology)

Expected Output:

Topology generated from subbase:
Topology(
  Space: [1, 2, 3],
  Collection of Subsets:
    [1, 2, 3]
    []
    [1]
    [2, 3]
  Properties: Not Connected, Compact, Separable
)

Explanation: By setting generate=True, the topology is automatically generated by closing the provided collection of subsets under finite unions and intersections, thus creating the minimal topology that contains the subbase.

More Advanced Usage

For a more in-depth exploration, the Jupyter notebooks in the GitHub repository offer guided examples on topics like:

• Constructing topologies based on equivalence relations.
• Exploring the interaction between topologies and continuous functions.
• Understanding the behavior of dense sets in topological spaces.
• Visualizing bases for topologies and how they define open sets.

Documentation

The complete documentation, including class details and function usage, is generated via Sphinx and hosted on GitHub Pages: https://nand0san.github.io/Topology/

Contributing

Feel free to contribute! Open an issue or submit a pull request for improvements, bug fixes, or new features.

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