# Space Crate
A Rust library providing mathematical structures and operations for topological spaces, simplicial complexes, and graphs. This crate is designed to support computational topology and geometry applications.
## Features
- **Topological Spaces**: Implementation of fundamental topological concepts
- Sets with basic operations (union, intersection, difference)
- Topological spaces with neighborhoods and open sets
- Metric spaces with distance functions
- Normed and inner product spaces
- **Simplicial Complexes**: Tools for working with simplicial complexes
- Simplex representation (points, edges, triangles, etc.)
- Chain complexes with boundary operations
- Support for arbitrary coefficient rings
- **Graph Theory**: Flexible graph data structures
- Support for both directed and undirected graphs
- Basic graph operations and set operations
- Vertex and edge point representation
## Usage
### Topological Spaces
```rust
use space::definitions::{Set, TopologicalSpace, MetricSpace};
// Define your own space type
struct MySpace {
// ... implementation details
}
impl Set for MySpace {
type Point = MyPoint;
// ... implement set operations
}
impl TopologicalSpace for MySpace {
type Point = MyPoint;
type OpenSet = MyOpenSet;
// ... implement topological operations
}
```
### Simplicial Complexes
```rust
use space::simplicial::{Simplex, SimplicialComplex, Chain};
// Create a simplex (e.g., a triangle)
let triangle = Simplex::new(2, vec![0, 1, 2]);
// Create a simplicial complex
let mut complex = SimplicialComplex::new();
complex.join_simplex(triangle);
// Compute boundaries
let boundary = complex.boundary::<i32>(2);
```
### Graphs
```rust
use space::graph::{Graph, Undirected};
use std::collections::HashSet;
// Create a graph
let mut vertices = HashSet::new();
vertices.insert(1);
vertices.insert(2);
let mut edges = HashSet::new();
edges.insert((1, 2));
let graph: Graph<_, Undirected> = Graph::new(vertices, edges);
```
## Dependencies
- `itertools`: For combinatorial operations
- `num`: For numeric traits and operations
## Examples
### Creating a Simplicial Complex
```rust
use space::simplicial::{Simplex, SimplicialComplex};
// Create a tetrahedron
let mut complex = SimplicialComplex::new();
complex.join_simplex(Simplex::new(3, vec![0, 1, 2, 3]));
```
### Working with Graphs
```rust
use space::graph::{Graph, Directed};
use std::collections::HashSet;
// Create a directed graph
let vertices: HashSet<_> = [1, 2, 3].into_iter().collect();
let edges: HashSet<_> = [(1, 2), (2, 3)].into_iter().collect();
let graph: Graph<_, Directed> = Graph::new(vertices, edges);
```