Science Fair Project Encyclopedia
Direct sum topology
In the mathematical field of topology a direct sum, direct disjoint sum or coproduct is an important universal construction for topological spaces. The canonical topology on the newly constructed space is called direct sum topology.
Given two topological spaces (X1,τ1) and (X2,τ2) we call
the disjunct set union of X1 and X2.
are called canonical injections.
The direct sum of two topological spaces is defined as
- (X1,τ1) + (X2,τ2): = (X1 + X2,τ1 + 2)
with the direct sum topology τ1 + 2 defined as
The direct sum topology is the finest topology such that the canonical injections are continuous.
Preservation of topological properties
- the direct sum of two topological spaces is disconnected
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details