# Réponse finale obtenue du bot:
response = « What a fascinating topic! Let’s dive into the properties of geometry, exploring both Euclidean and non-Euclidean geometries, trigonometry, and topology.
**Euclidean Geometry**
In Euclidean geometry, we deal with the familiar world where:
1. **Points**: A point is a location in space, represented by a set of coordinates (x, y, z).
2. **Lines**: A line is a set of points that extend infinitely in two directions.
3. **Planes**: A plane is a flat surface that extends infinitely in all directions.
4. **Angles**: Angles are formed by the intersection of two lines or planes.
5. **Distance**: Distance is measured using the Pythagorean theorem (a² + b² = c²).
6. **Similarity**: Similar figures have proportional corresponding sides and equal corresponding angles.
**Non-Euclidean Geometry**
In non-Euclidean geometry, we venture into unfamiliar territories where:
1. **Curved spaces**: Spaces with constant positive curvature (spherical) or negative curvature (hyperbolic) deviate from Euclid’s parallel postulate.
2. **Hyperbolic geometry**: In hyperbolic space, lines appear to diverge as they approach infinity, and angles are not always additive.
3. **Spherical geometry**: On the surface of a sphere, great circles (circles that pass through the center) have constant positive curvature.
**Trigonometry**
In trigonometry, we explore relationships between the sides and angles of triangles:
1. **Angles**: Angles can be measured in degrees or radians.
2. **Sines**, **cosines**, and **tangents**: These functions relate to ratios of side lengths and angles.
3. **Triangles**: Triangles have three sides ( legs ) and three angles, with relationships between them.
**Topology**
In topology, we study the properties that remain unchanged under continuous deformations:
1. **Connectedness**: A space is connected if it cannot be divided into separate components.
2. **Compactness**: A space is compact if it is closed and bounded.
3. **Hole-iness**: A topological space can have holes or cavities, affecting its connectivity.
Some key concepts in topology include:
1. **Homotopy**: Two curves are homotopic if they can be continuously deformed into each other without cutting or pasting.
2. **Homeomorphism**: Two spaces are homeomorphic if there exists a continuous bijective function between them.
This is just a brief introduction to the fascinating world of geometry, trigonometry, and topology! There’s much more to explore and discover in these fields. »