Comparison of topologies
An order relation on the set of all topologies on one and the same set . A topology
majorizes a topology
(or
is not weaker than
), if the identity mapping
, where
is the set
with the topology
,
, is continuous. Moreover, if
, then
is stronger than
(or
is weaker than
).
The following statements are equivalent:
1) majorizes
.
2) For any , every neighbourhood of
in the topology
is a neighbourhood of
in the topology
.
3) For any , the closure of
in
contains the closure of
in
.
4) Every set from , closed in
, is also closed in
.
5) Every set that is open in is open in
.
In the ordered set of topologies on , the discrete topology is the strongest, while the topology whose only closed sets are
and
is the weakest. Figuratively speaking, the stronger the topology, the more open sets, closed sets and neighbourhoods there are in
; the stronger the topology, the smaller the closure of a set (and the larger its interior) and the smaller the number of everywhere-dense sets.
Comparison of topologies. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Comparison_of_topologies&oldid=11528