definition of connected topological space
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 connected topological space.
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:
\(*T\): \(\in \{\text{ the topological spaces }\}\)
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Conditions:
\(\lnot \exists U_1, U_2 \in \{\text{ the nonempty open subsets of } T\} (T = U_1 \cup U_2 \land U_1 \cap U_2 = \emptyset)\)
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2: Note
'Connected' or 'disconnected' is always as a topological space, not as a subset of a topological space, which means that when a subset is said to be connected or disconnected, it is as the topological subspace, which means that it is not about whether the subset is the union of some nonempty disjoint open subsets of the ambient topological space, but about whether the topological subspace is the union of some nonempty disjoint open subsets of the topological subspace.
For example, let \(T' = \mathbb{R}\) and \(T = [-2, -1] \cup [1, 2] \subset T'\). Then, \(T\) is not connected although it is not any union of some nonempty disjoint open subsets of \(T'\): it cannot be, because \(T\) is not open on \(T'\). \(T\) is not connected because \([-2, -1]\) and \([1, 2]\) are some nonempty disjoint open subsets of \(T\).
