[Math] show that $[0,1]$ is not compacts as a subspaces of $R_K$

general-topology

  1. Recall that $R_K$ denotes the R in the $K -topology$.

a) Show that $[0,1]$ is not compacts as a subspaces of $R_K$

i know that $R_K $ is finer then R since its basis contain the basis of $R$

$R_K =(a,b) – \frac{1}{n}$ which is not open because $\frac{1}{n} $ is not closed as complement of open set is closed

Best Answer

Recall that the K-topology is generated by the basis {$(a, b), (a, b) − K | a < b$}, where

K = { $\frac{1}{n}| n\in Z^+$} for each i ,let $U_i = (\frac{1}{i} ,2) U (-1,1) - k:$ the open cover {${u_i}$} of $[0,1]$ does not have finite subcover so$ [0,1]$ is not compacts

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