| Finer Topology |
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| CATEGORIES ABOUT COMPARISON OF TOPOLOGIES | |
| general topology | |
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DEFINITION Let τ1 and τ2 be two topologies on a set ''X'' such that τ1 is Contained In τ2: :. That is, every set open under τ1 is also open under τ2. Then the topology τ1 is said to be a coarser ('''weaker''' or '''smaller''') '''topology''' than τ2, and τ2 is said to be a '''finer''' ('''stronger''' or '''larger''') '''topology''' than τ1. If additionally : we say τ1 is strictly coarser than τ2 and τ2 is '''strictly finer''' than τ1. The Binary Relation ⊆ defines a Partial Ordering Relation on the set of all possible topologies on ''X''. N.B. There are some authors, especially Analyst s, who use the terms ''weak'' and ''strong'' with opposite meaning. EXAMPLES The finest topology on ''X'' is the Discrete Topology . The coarsest topology on ''X'' is the Trivial Topology . In Function Space s and spaces of Measure s there are often a number of possible topologies. See Topologies On The Set Of Operators On A Hilbert Space for some intricate relationships. All possible Polar Topologies on a Dual Pair are finer than the Weak Topology and coarser than the Strong Topology . PROPERTIES Let τ1 and τ2 be two topologies on a set ''X''. Then the following statements are equivalent:
Two immediate corollaries of this statement are
One can also compare topologies using Neighborhood Base s. Let τ1 and τ2 be two topologies on a set ''X'' and let ''B''''i''(''x'') be a local base for the topology τ''i'' at ''x'' ∈ ''X'' for ''i'' = 1,2. Then τ1 ⊆ τ2 if and only if for all ''x'' ∈ ''X'', each open set ''U''1 in ''B''1(''x'') contains some open set ''U''2 in ''B''2(''x''). Intuitively, this makes sense: a finer topology should have smaller neighborhoods. LATTICE OF TOPOLOGIES The set of all topologies on a set ''X'' together with the partial ordering relation ⊆ forms a Complete Lattice . That is, any collection of topologies on ''X'' have a ''meet'' (or Infimum ) and a ''join'' (or Supremum ). The meet of a collection of topologies is the Intersection of those topologies. The join, however, is not generally the Union of those topologies (the union of two topologies need not be a topology) but rather the topology Generated By the union. Every complete lattice is also a Bounded Lattice , which is to say that is has a Greatest and Least Element . In the case of topologies, the greatest element is the Discrete Topology and the least element is the Trivial Topology . SEE ALSO
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