description/proof of that for open surjective continuous map between topological spaces, image of basis of domain is basis of codomain
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Proof
- 3: Note 1
- 4: Note 2
- 5: Note 3
Starting Context
- The reader knows a definition of continuous map.
- The reader knows a definition of open map.
- The reader knows a definition of surjection.
- The reader knows a definition of basis of topological space.
- The reader admits the proposition that for any map between sets, the composition of the map after any preimage is contained in the argument set.
Target Context
- The reader will have a description and a proof of the proposition that for any open surjective continuous map between any topological spaces, the image of any basis of the domain is a basis of the codomain.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
//
Statements:
//
2: Proof
Whole Strategy: Step 1: see that each
Step 1:
Each
Step 2:
For any point,
There is a point,
There is a
So, yes.
3: Note 1
So, if
4: Note 2
The requirements for
5: Note 3
As a quotient map is not necessarily open, this proposition cannot be applied to general quotient maps, so, a quotient space of a 2nd-countable topological space is not guaranteed to be 2nd-countable by this proposition, and in fact, there are some non-2nd-countable quotient spaces of 2nd-countable topological spaces.