definition of convergence of map from topological space minus point into topological space w.r.t. point
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of topological space.
Target Context
- The reader will have a definition of convergence of map from topological space minus point into topological space with respect to 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:
\( T_1\): \(\in \{\text{ the topological spaces }\}\)
\( T_2\): \(\in \{\text{ the topological spaces }\}\)
\( t_1\): \(\in T_1\)
\( f\): \(: T_1 \setminus \{t_1\} \to T_2\)
\(*t_2\): \(\in T_2\)
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Conditions:
\(\forall N_{t_2} \subseteq T_2 \in \{\text{ the neighborhoods of } t_2\} (\exists N_{t_1} \subseteq T_1 \in \{\text{ the neighborhoods of } t_1\} (f (N_{t_1} \setminus \{t_1\}) \subseteq N_{t_2}))\)
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There may be some multiple convergences, but when there is the unique convergence, the convergence can be denoted as \(lim_{t' \to t_1} f (t')\).
2: Note
The intention of taking \(T_1 \setminus \{t_1\}\) instead of \(T_1\) as the domain of \(f\) is that \(f\) is not required to be defined at \(t_1\), not that "\(f\) must not be defined at \(t_1\)".
In fact, when there is an \(f': T_1 \to T_2\), \(f: T_1 \setminus \{t_1\} \to T_2\) can be canonically defined as the domain restriction of \(f'\), and the convergence of \(f'\) with respect to \(t_1\) can be regarded to be the convergence of \(f\) with respect to \(t_1\).
The reason why we need to think of \(T_1 \setminus \{t_1\}\) is that there are some important cases in which \(f\) is not defined at \(t_1\): for example, let \(T_1 = (- \delta, \delta) \subseteq \mathbb{R}\), \(T_2 \in \{\text{ the finite-dimensional } \mathbb{R} \text{ vectors spaces }\}\) with the canonical topology, \(t_1 = 0\), and for a \(g: T_1 \to T_2\), \(f: (- \delta, \delta) \setminus \{0\} \to T_2, t \mapsto (g (t) - g (0)) / (t - 0)\), where \(lim_{t \to 0} f (t)\) is the derivative.
When \(T_2\) is Hausdorff and \(\{t_1\} \subseteq T_1\) is not open, there can be only 1 convergence: let us suppose that there were 2 convergences, \(t_2, t'_2 \in T_2\), such that \(t_2 \neq t'_2\); there would be some open neighborhoods of \(t_2\) and \(t'_2\), \(U_{t_2}\) and \(U_{t'_2}\), such that \(U_{t_2} \cap U_{t'_2} = \emptyset\); there would be some neighborhoods of \(t_1\), \(N_{t_1}\) and \(N'_{t_1}\), such that \(f (N_{t_1} \setminus \{t_1\}) \subseteq U_{t_2}\) and \(f (N'_{t_1} \setminus \{t_1\}) \subseteq U_{t'_2}\); but \(f ((N_{t_1} \cap N'_{t_1}) \setminus \{t_1\}) \subseteq U_{t_2} \cap U_{t'_2} = \emptyset\), which would mean that \(N_{t_1} \cap N'_{t_1} = \{t_1\}\), which would mean that \(\{t_1\}\) was open, a contradiction against the supposition.
In fact, when \(\{t_1\} \subseteq T_1\) is open, each point, \(t_2 \in T_2\), is a convergence with respect to \(t_1\), because for any \(N_{t_2}\), \(N_{t_1} := \{t_1\}\) can be chosen.
