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Let $T$ be a topology on $\mathbb{R}$ for which $[a,b)$ form a basis, $a \lt b$, let $S$ be a topology on $\mathbb{R}$ s.t. $T$ is contained in $S$, then which of the following are true and how?

a) Either $S = T$ or $S$ is discrete topology.
b) If, moreover, the map, $x$ going to $-x$ is continuous for $S$, then $S$ is discrete.
c) If, moreover, the map, $x$ going to $-x$ is a homeomorphism for $S$, then $S$ is discrete.
d) If, moreover, the map, $x$ going to modulus of $x$ is a homeomorphism for $S$, then $S$ is discrete.

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Please do not post only a question with no effort from yourself. –  Austin Mohr Feb 1 '13 at 17:31
As the lower limit topology isn't discrete, you also presumably have a typo in a) ("or $S$ is discrete"). –  gnometorule Feb 1 '13 at 17:37
@Amr i am done with c) and its coming out to be true, and for the same reason d) is true, a) is definitely false. so basically i am confused with b only. –  anonymous Feb 1 '13 at 17:38
@anonymous The remark tha t$S$ and $T$ should be switched seems to apply to all parts of the problem –  Hagen von Eitzen Feb 1 '13 at 17:46
Hint: If $x\mapsto -x$ is $S$-continuous, then intervals of the form $(a,b]$ will be $S$-open. –  Harald Hanche-Olsen Feb 1 '13 at 18:39

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