The discussion focuses on identifying open sets within the metric space \( (E=]-1,0]\cup\{1\}, d) \), where the metric \( d \) is defined as \( d(x,y)=|x-y| \). Participants emphasize the importance of defining open sets in the context of metric spaces, specifically using the standard topology induced by the absolute value metric. The explicit open sets for this space include intervals around points in \( E \) that do not include the endpoints of \( ]-1,0] \) and the isolated point \( 1 \).
PREREQUISITESStudents and professionals in mathematics, particularly those studying topology and metric spaces, as well as educators looking to deepen their understanding of open sets in metric spaces.
Can you please post your progress on this question or anything you have tried? Start with defining open sets in an arbitrary metric space.eraldcoil said:Let $$(E=]-1,0]\cup\left\{1\right\},d) $$ metric space with $$d$$ metric given by $$d(x,y)=|x-y|$$, and $$||$$absolute value.
How I can find open sets of E explicitly?
Thanks in advance.