Concept
Topological space
Overview
In mathematics, a topological space is, roughly speaking, a space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets.
A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds.
Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spaces in their own right is called general topology (or point-set topology).
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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This property of quantities, according to which no part of them is the smallest possible (no part simple), is called their continuity. Space and time are quanta continua, because no part of them can be given, without enclosing it within boundaries (points and moments), consequently, this given part is itself a space or a time. Space, therefore, consists only of spaces, and time of times. Points and moments are only…
Immanuel Kant, The Critique of Pure Reason
States what Topological space is.
Two neighbouring events have a measurable quantitative relation called “interval”, which is sometimes analogous to distance in space, sometimes to lapse of time. In the former case it is called space-like, in the latter time-like. The interval between two events is time-like when one body might be present at both--for example, when both are parts of the history of your body. The interval is space-like in the…
Bertrand Russell, Philosophy
States what Topological space is.
We may, if we please, call the proposition, “Three is two and one,” a definition of the number three, and assert that arithmetic, as it has been asserted that geometry, is a science founded on definitions. But they are definitions in the geometrical sense, not the logical; asserting not the meaning of a term only, but along with it an observed matter of fact. The proposition, “A circle is a figure bounded by a line…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Topological space is.
Related ideas
- Felix Hausdorff
- Finite set
- Function (mathematics)
- Mathematics
- Real number
- Manifold
- Number theory
- Intersection (set theory)
- Power set
- Limit (mathematics)
- Set (mathematics)
- Subset
- Morphism
- Theory of computation
- Bijection
- Union (set theory)
- Henri Poincaré
- Complement (set theory)
- Complex number
- Category theory
- Carl Friedrich Gauss
- Equivalence relation
- Axiom
- Empty set
Place in the tradition
Part of Modal logic.