I am having some trouble understanding what a topology means:

The definition says that: if $X$ is a non-empty set then a set $T$ of subsets of $X$ is said to be a topology on $X$ if

(i) $X$ and the empty set,$\emptyset$, belong to $T$

(ii) The union of any (finite or infinite ) number of sets in $T$ belongs to $T$

(iii)The intersection of of any two sets in $T$ belongs to $T$

Okay then if I define the sets $T_1$ and $T_2$ to be:


$T_2=\{]-\infty,2[ \cup ]3,+\infty[,[2,3],\emptyset,R\}$

Then I guess that $T_1$ and $T_2$ must be two topologies on the set of real numbers $R$ as they satisfy the above three conditions.

Another definition says that the members of a topology are said to be open sets, so I want to ask if the set $[2,3]$ is considered to be open in the topological space ($R$,$T_1$)? and is this same set $[2,3]$ considered to be both open and closed in the toplological space ($R$,$T_2$) as it is present with its complement in $T_2$? and what about the subset $[5,7]$ of $R$ is it considered to be neither open nor closed in the above two topological spaces??

Any help is appreciated.

  • 1
    $\begingroup$ The answer is yes to all your questions. $\endgroup$ – saulspatz Jan 20 '18 at 8:55

1) $[2,3]\in\tau_{1}$, so $[2,3]$ is open in $\tau_{1}$.

2) $[2,3]\in\tau_{2}$, so $[2,3]$ is open in $\tau_{2}$. Now $[2,3]^{c}=(-\infty,2)\cup(3,\infty)\in\tau_{2}$, so $[2,3]^{c}$ is open in $\tau_{2}$, so $[2,3]$ is closed in $\tau_{2}$.

3) $[5,7]\notin\tau_{1},\tau_{2}$, so $[5,7]$ is neither open in $\tau_{1}$ nor in $\tau_{2}$. $[5,7]^{c}=(-\infty,5)\cup(7,\infty)\notin\tau_{1},\tau_{2}$, so $[5,7]$ is neither closed in $\tau_{1}$ nor in $\tau_{2}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.