The trivial topology is the topology with the least possible number of . Although it has many other useful properties, these do not make up for this one failing.
Other properties of an indiscrete space ''X''—many of which are quite unusual—include:
- The only Closed Set s are the empty set and ''X''.
- The only possible Basis of ''X'' is {''X''}.
- Because ''X'' is not T0 , it does not satisfy any of the higher T Axioms either. In particular, it is not a Hausdorff Space .
- ''X'' is, however, Regular , Completely Regular , Normal , and Completely Normal ; all in a rather vacuous way though, since the only closed sets are ∅ and ''X''.
- Not being Hausdorff, ''X'' is not an Order Topology , nor is it Metrizable .
- ''X'' is Compact and therefore Paracompact , Lindelöf , and Locally Compact .
- If a Function has ''X'' as its Range , it is Continuous .
- ''X'' is Path-connected and so Connected .
- ''X'' is First-countable , Second-countable , and Separable .
- All Subspace s of ''X'' have the trivial topology.
- All Quotient Space s of ''X'' have the trivial topology
- Arbitrary Product s of trivial topological spaces, with either the Product Topology or Box Topology , have the trivial topology.
- All Sequence s in ''X'' Converge to every point of ''X''. In particular, every sequence has a convergent subsequence (the whole sequence).
- The Interior of every set except ''X'' is empty.
- The , a property that characterizes trivial topological spaces.
- If ''S'' is any subset of ''X'' with more than one element, then all elements of ''X'' are Limit Point s of ''S''. If ''S'' is a Singleton , then every point of ''X'' \ ''S'' is still a limit point of ''S''.
- ''X'' is a Baire Space .
- Two topological spaces carrying the trivial topology are Homeomorphic Iff they have the same Cardinality .
In some sense the opposite of the trivial topology is the Discrete Topology , in which every subset is open.
The trivial topology belongs to a Pseudometric Space in which the distance between any two points is Zero , and to a Uniform Space in which the whole cartesian product ''X'' × ''X'' is the only entourage.
Let be the on a given set is ''left Adjoint'' to ''F''.)
- Lynn Arthur Steen and J. Arthur Seebach, Jr., '' Counterexamples In Topology '', (1978) Dover Publications, ISBN 0-486-68735-X. ''(See example 4)''
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