Hi, this is a topology question, where the foot notation indicates a type of top

Question

Hi, this is a topology question, where the foot notation indicates a type of topology, e.g. upper topology, standard topology, etc. If you need to double check these topology notations, please refer to the post: https://www.chegg.com/homework-help/questions-and-answers/hi-topology-problem-please-justify-inclusions-topologies-explain-strict-thank-advance-help-q26571950

Some given hints: Tcodisc <Tfin / Tup < Tst < Tk < Tuplim < Tdisc, with the "<" meaning "is in". With these relation, all we need to find is a break point and the other topology to the left/right is automatically justified.

1. Consider the topologies Tdisc, Tcodisc, Tfin, Tcount, Tst, TK, Tup, Tuplim on R. For each topology, (i) say if K := { l . (ii) say if K U (0) is closed (ii) say if R is Hausdorff (iv) say if R is connected, except for the topology Tk. You can assume that (R, Tst) is n Nt is closed n+1 connected, which will be proven in class. (v) say if Q, endowed with the subspace topology, is connected. Justify each answer. To make the exercise shorter, it is useful to keep in mind that these topologies are related, X} is 2. Prove that a space (XT) is Hausdorff if and only if the diagonal := {(x, x) 1 x closed in (XxX,T)

Explanation / Answer

2.  Suppose X is Hausdorff. The diagonal is closed if and only if the complement c = X × X is open. Let a × b c , i.e. a and b are distinct points in X. Since X is Hausdorff there exists open sets U and V in X such that a U, b V and U V = . Hence a × b U × V and U × V open in X × X. Furthermore (U × V ) = , since U and V are disjoint. So for every point a × b c there exists an open set Ua×b such that a × b Ua×b c .

(By Ex. 13.1 of the Topology book by James Munkres) it follows that c open, i.e. closed. Now suppose is closed. If a and b are two distinct points in X then a × b c . Since c is open there exists a basis element U × V , U and V open in X, for the product topology, such that a × b U × V c . Since U × V c it follows that U V = . Hence U and V are open sets such that a U, b V and U V = , i.e. X is Hausdorff.

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