What are Some Examples of Topologically Equivalent Sets and Their Properties?
Topologically equivalent sets are sets that can be continuously deformed into one another without tearing, gluing, or intersecting. In other words, they have the same shape and dimensionality.
Examples of Topologically Equivalent Sets:
- A circle and an ellipse
- A square and a rectangle
- A coffee cup and a mug
- A sphere and a ball
- A torus and a pretzel
Properties of Topologically Equivalent Sets:
- They have the same number of holes.
- They have the same number of boundary components.
- They can be continuously mapped onto one another in a one-to-one correspondence.
- They have the same Betti numbers.
- They have the same fundamental group.
Related Questions and Answers
- What is the difference between topological equivalence and homeomorphism? Homeomorphism is a stronger condition than topological equivalence that requires a continuous bijection between the sets.
- Can two sets with different cardinalities be topologically equivalent? No, they cannot.
- Can a set be topologically equivalent to itself? Yes, a set is always topologically equivalent to itself.
- What is the topological equivalence of a sphere with n holes? A sphere with n holes is topologically equivalent to a sphere with n + 1 handles.
- What is the topological equivalence of a torus with n holes? A torus with n holes is topologically equivalent to the product of n circles.
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