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Definition: Topological Space, Topology
A topological space is an ordered pair $(X,\mathcal O)$ of a set $X$ together with a set of its subsets subset $\mathcal {O}$ (called its topology), which obeys the following axioms:
 The empty set \(\emptyset\) and the whole set \(X\) are elements of the topology (i.e. \({\emptyset\in\mathcal {O}}\) and \({X\in\mathcal {O}}\) ).
 The intersection of finitely many elements of the topology $\mathcal O$ is an element of the topology, i.e. if \(U_{1},\ldots ,U_{n}\in {\mathcal {O}}\), then also \(U_{1}\cap \ldots \cap U_{n}\in {\mathcal {O}}\).
 Every union of elements of the topology is again an element of it, i.e. with \(U_{i}\in {\mathcal {O}}\) for each \(i\in I\) (for an arbitrary index set \(I\)) we have also \(\bigcup _{i\in I}U_{i}\in {\mathcal {O}}\).
Notes
 A topological space is defined simply by a postulating that subsets of an arbitrary carrier set $X$ fulfill the above properties.
 Because the axioms $1$ to $3$ do not uniquely define a topological space, it is possible to define many different topologies $\mathcal O$ on a set $X.$
 Even a set $X=\{a,b\}$ with only two elements has different topologies (resulting in different topological spaces). How many? The reader might use this example as an exercise. (Hint: Consider all nonempty subsets of the power set $\mathcal P(\{a,b\})$).
Mentioned in:
Axioms: 1
Chapters: 2 3 4
Definitions: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Examples: 32
Parts: 33 34
Proofs: 35 36 37 38 39 40 41 42 43 44 45 46
Propositions: 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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References
Adapted from CC BYSA 3.0 Sources:
 Brenner, Prof. Dr. rer. nat., Holger: Various courses at the University of Osnabrück