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topological space∗
djao†
2013-03-21 12:32:37
A topological space is a set X together with a set T whose elements are
subsets of X, such that
• ∅∈T
• X∈T
• If Uj ∈ T for all j ∈ J, then
S
j∈J
Uj ∈ T
• If U ∈ T and V ∈ T , then U ∩ V ∈ T
Elements of T are called open sets of X. The set T is called a topology on
X. A subset C ⊂ X is called a closed set if the complement X \ C is an open
set.
A topology T 0 is said to be finer (respectively, coarser) than T if T 0 ⊃ T
(respectively, T 0 ⊂ T ).
Examples
• The discrete topology is the topology T = P(X) on X, where P(X) denotes
the power set of X. This is the largest, or finest, possible topology on X.
• The indiscrete topology is the topology T = {∅, X}. It is the smallest or
coarsest possible topology on X.
• Subspace topology
• Product topology
• Metric topology
∗ hTopologicalSpacei
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References
[1] J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
[2] J. Munkres, Topology (2nd edition), Prentice Hall, 1999.
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