description/proof of that constant map between topological spaces is continuous
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of continuous, topological spaces map.
Target Context
- The reader will have a description and a proof of the proposition that any constant map between any topological spaces is continuous.
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:
\(T_1\): \(\in \{\text{ the topological spaces }\}\)
\(T_2\): \(\in \{\text{ the topological spaces }\}\)
\(f\): \(: T_1 \to T_2\)
//
Statements:
\(\exists t_0 \in T_2 (\forall t \in T_1 (f (t) = t_0))\)
\(\implies\)
\(f \in \{\text{ the continuous maps }\}\)
//
2: Proof
Whole Strategy: Step 1: see that for each \(t \in T_1\), \(f\) is continuous at \(t\).
Step 1:
Let \(t \in T_1\) be any.
\(f (t) = t_0\).
Let \(U_{t_0} \subseteq T_2\) be any open neighborhood of \(t_0\) on \(T_2\).
\(T_1\) is an open neighborhood of \(t\) on \(T_1\) and \(f (T_1) = \{t_0\} \subseteq U_{t_0}\).
So, \(f\) is continuous at \(t\).
As \(t\) is arbitrary, \(f\) is continuous.
