Answer to Question #282264 in Differential Geometry | Topology for Nemar

Question #282264

"\\text { Find the limit points of }\\left\\{\\frac{1}{n} \\mid n \\in \\mathbb{N}\\right\\} \\text { in } \\mathbb{R} \\text {. }" (DG)


1
Expert's answer
2021-12-27T18:32:37-0500

A limit point of a set "A" in a topological space "X" is a point "x" that can be "approximated" by points of "A" in the sense that every neighbourhood of "x" with respect to the topology on "X" also contains a point of "A" other than "x" itself.

In our case, for the set "A=\\{\\frac{1}n\u2223n\u2208\\N\\}" in "\\R" with standard topoogy the limit point is "x=0." Indeed, any basic open neighbourhood "(-\\varepsilon,\\varepsilon)" of "x=0" contains all elements "x_n=\\frac{1}n" for natural numbers "n>\\frac{1}{\\varepsilon}."


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