Topology : Topology
Study concepts, example questions & explanations for Topology
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Example Questions
Example Question #1 : Euler Characteristic
What is the Euler Characteristic for a triangle?
The Euler Characteristic is calculated by the following equation.
This is also written as,
Recall that the Betti numbers represent the vertices, edges, and faces of the object.
In this particular question the Betti numbers can be calculated therefore,
now substitute them into the formula to calculate the Euler Characteristic and solve.
Example Question #1 : Topological Spaces
Determine whether the statement is true or false:
The Betti number of an object gives the number of holes in each dimension in a given topological space.
False
True
True
In topology, Betti numbers represent the various dimensions of a topological space in regards to the number of holes that are present. There are four different simplices depending on the dimension: zero, one, two, three.
These are also referred to as:
Zero simplex refers to a point or vertex.
One simplex refers to a line or edge (which is connected by two points)
Two simplex refers to a face (which is connected by three lines)
Three simplex refers to the 3D object created by two faces.
Therefore by definition the statement,
"The Betti number of an object gives the number of holes in each dimension in a given topological space." is true.
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