MHB About open sets in a metric space.

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To find open sets in the metric space \( E = ]-1,0] \cup \{1\} \) with the metric \( d(x,y) = |x-y| \), one must first understand the definition of open sets in a metric space. An open set around a point \( x \) includes all points \( y \) such that the distance \( d(x,y) \) is less than some radius \( r \). In this case, open sets can be identified by examining intervals around points in \( E \) that do not include the endpoints. The intervals \( (-1, 0) \) and \( (1 - r, 1 + r) \) for small \( r \) can be considered, but care must be taken with the endpoints of \( E \). Ultimately, the open sets in this metric space will be combinations of these intervals, excluding the endpoints.
eraldcoil1
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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.
 
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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.
Can you please post your progress on this question or anything you have tried? Start with defining open sets in an arbitrary metric space.
 

Thread 'About the definition of topological manifold using closed sets'

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in ##\mathbb R^n##. It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on ##\mathbb R^n## ? I'm positive for this. Perhaps the definition of smooth manifold would be problematic, though.
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