Families of Shapes: Defining Topology in Topology

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However, there is no strict definition of a family of shapes in topology and it can vary depending on the specific requirements of a study or model.
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SW VandeCarr
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Families of "shapes"

Is there a general definition of a 'family' in topology or can a family be defined to suit one's specific requirements?

I'm thinking in reference to the Calabi Yau (CY) family of shapes which are used as a model in most string theories. CY shapes are described as a family and they have anywhere from one to over 400 holes. Given that topological genera are defined more or less by the number of holes in a manifold, what more general topological characteristic would allow for the definition of a family?
 
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A family of shapes is generally defined as a set of shapes that have similar topological characteristics. These characteristics can include the number of holes, the type of surface, or the local curvature. For instance, the Calabi Yau family is defined by the fact that all of its shapes have the same number of complex dimensions (3) and the same number of holes (at least one). It is also defined by its local curvature which is negative, meaning it can be bent inwards to form a kind of “dome” shape.
 

Related to Families of Shapes: Defining Topology in Topology

1. What is topology and how does it relate to families of shapes?

Topology is a branch of mathematics that studies the properties of space that are preserved under continuous deformations, such as stretching, twisting, and bending. It is closely related to families of shapes because topology focuses on the global properties of shapes, rather than their specific geometric details.

2. What is the importance of defining topology in topology?

Defining topology is crucial in topology because it allows us to distinguish between different shapes and understand their fundamental properties. By defining a topology on a shape, we can determine whether it is connected, compact, or has holes, which are important characteristics in the study of topology.

3. How do we define a topology on a family of shapes?

To define a topology on a family of shapes, we need to specify a set of rules, known as the axioms of topology, that describe how points in the shape can be connected to each other. This includes defining open sets, which are subsets of the shape that satisfy certain properties, and how these open sets can be combined to form new open sets. This process allows us to define a topology on a shape in a consistent and rigorous manner.

4. What are some common topologies used in the study of families of shapes?

There are several common topologies used in the study of families of shapes, including the discrete topology, the Euclidean topology, and the Zariski topology. The discrete topology is the simplest topology, where every subset of a shape is considered open. The Euclidean topology is based on the familiar notion of distance in Euclidean space, while the Zariski topology is used in algebraic geometry to study the solutions of polynomial equations.

5. How does topology help us understand the structure of families of shapes?

Topology provides a powerful tool for understanding the structure of families of shapes. By focusing on the global properties of shapes and their relationships, rather than their specific geometric details, topology allows us to study the fundamental structure of shapes and identify important characteristics, such as connectedness, compactness, and dimension.

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