description/proof of that for map between topological spaces and domain point, if there are superspaces of domain and codomain, open neighborhoods of point and of point image on superspaces, and continuous map from domain neighborhood into codomain neighborhood that is restricted to original map on intersection of domain neighborhood and original domain, original map is continuous at point
Topics
About: topological space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
- 3: Proof
- 4: Note
Starting Context
- The reader knows a definition of topological space.
- The reader knows a definition of continuous map at point.
- The reader admits the proposition that for any topological space and any sub topological space that is open on the base space, any subset of the subspace is open on the subspace if and only if it is open on the base space.
Target Context
- The reader will have a description and a proof of the proposition that for any map between any topological spaces and any domain point, if there are some superspaces of the domain and the codomain, some open neighborhoods of the point and of the point image on the superspaces, and a continuous map from the domain neighborhood into the codomain neighborhood that is restricted to the original map on the intersection of the domain neighborhood and the original domain, the original map is continuous at the point.
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:
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Statements:
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2: Natural Language Description
For any topological spaces,
3: Proof
Let us suppose that there are such
For any open neighborhood,
In fact,
So, for each neighborhood,
4: Note
The superspaces are usually taken to be some
Note that the subspaces do not necessarily have any chart, because a subspace is not necessarily a
Bringing in an arbitrary in-norm-sense continuous function between the topological subspaces that (the function) is not based on any superspaces should not guarantee the continuousness of the original map in topological sense, unless proved otherwise.