And the induced topology agrees with the product topology.
2.
Proximity maps will be continuous between the induced topologies.
3.
The metric identification preserves the induced topologies.
4.
The induced topology is the indiscrete topology.
5.
The induced topology is the original topology.
6.
The Helly space has a topology; namely the induced topology as a subset of " I I ".
7.
Now it is obvious that if a system is weaker than another, the induced topology is weaker . . but what about the converse?
8.
One can show that the compact subsets of " X " c and " X " coincide and the induced topologies are the same.
9.
Because any " G " ? subset of a Polish space is again a Polish space, the theorem also shows that any " G " ? subset of a Polish space is the union of a countable set and a set that is perfect with respect to the induced topology.
10.
In general the group \ Gamma \ cap \ overline H is equal to the congruence closure of H in \ Gamma, and the congruence topology on \ Gamma is the induced topology as a subgroup of \ mathbf G ( \ mathbb A _ f ), in particular the congruence completion \ overline \ Gamma is its closure in that group.
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