[topology-subspace]
Subtopology := let . Inherit net system
Equivalently defined, the subtopology is the smallest topology that makes the embedding map continuous
Inheritance of subtopology. is a subtopology <==> is a subtopology
Proof According to the associativity of +
[closed-in-subspace] closed Characterization of in subspace
Example
Indicates that may have limit point or , but the limit point of can only be
is a closed set
- ==>
- ==>
[quotient-topology]
[product-topology]
:= The smallest topology in which all component mappings are continuous, i.e., with the family of sets
to generate a net system of finite intersections
Because is a component mapping
The product of closed sets is also a closed set. by the definition of limit point and and logic
The image does not necessarily pass closed sets. Example The closed set mapped to the axis yields a non-closed set
[sum-topology]
The topology of is the largest topology that makes the embedding continuous
The point grid system of for all constitutes the point grid system of the sum space in the copy of the sum space, in which the form of the set is