A description/proof of that inclusion into topological space from subspace is continuous
Topics
About: topological space
The table of contents of this article
Starting Context
- The reader knows a definition of subspace topology.
- The reader knows a definition of continuous map.
Target Context
- The reader will have a description and a proof of the proposition that for any topological space, the inclusion from any subspace into the topological space 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: Description
For any topological space, \(T_1\), and any subspace, \(T_2 \subseteq T_1\), the inclusion, \(f: T_2 \rightarrow T_1\) is continuous.
2: Proof
For any open set, \(U \in T_1\), \(f^{-1} (U) = U \cap T_2\), which is open on \(T_2\), by the definition of subspace topology.
