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What is the exact definition of the dimension of a topological space?
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HallsofIvy said:?? Your title is "the definition of dimension" but your question is "what is the exact definition of a topological space?" Which is it? The definition of "dimension" depends strongly on exactly what kind of space you are dealing with. The definition of "topological space", however, is quite simple:
A topological space is a set with a topology!
And a topology (for set X) is a collection, T, of subsets of X satisfying:
The empty set is in T.
The entire set X is in T.
The union of any collection of sets in T is also in T.
The intersection of any finite collection of sets in T is also in T.
For any set X, whatsoever, the following are topologies on T:
The collection of all subsets of X. (Often called the "discrete" topology.)
The collection containing only the empty set and X. (Often called the "indiscrete" topology.)
Dimension refers to the measure of the size or extent of an object or space in terms of length, width, and height.
In the physical world, there are three dimensions: length, width, and height. However, in mathematics and theoretical physics, there are more than three dimensions, such as the fourth dimension of time.
2D refers to two-dimensional objects or spaces that only have length and width. 3D, on the other hand, refers to three-dimensional objects or spaces that have length, width, and height.
The fourth dimension is often referred to as time. In physics, it is used to describe the movement of objects through time and space.
Some theories in physics suggest the possibility of additional dimensions beyond the three we can perceive. However, these dimensions are not directly observable and are only understood through mathematical models.