Discussion
Which of the following spaces need not be Hausdorff ?
Answer: C
The space that need not be Hausdorff is an indiscrete topological space.
In the indiscrete topology the only open sets are the empty set and X. If X has two points, the only open set containing either of them is X itself, so they cannot be separated by disjoint open sets.
The others are always Hausdorff. A space is Hausdorff exactly when its diagonal \(\Delta\) is closed in \(X \times X\) (A). In a metric space two points at distance \(d\) lie in disjoint balls of radius \(d/2\) (B). And a product of Hausdorff spaces is Hausdorff, since points differing in one coordinate are separated by preimages of disjoint open sets (D).
No comment is present. Be the first to comment.
Loading…
Post your comment
Please log in or register to post a comment.
Examination Categories
- SSC Exams
- Banking Exams
- Teaching Exams
- Civil Services Exams
- Railways Exams
- Engineering Recruitment Exams
- Defence Exams
- State Government Exams
- Police Exams
- Insurance Exams
- Nursing Exams
- Other Government Exams
- NRA CET
- PG Entrance Exams
- Campus Placement
- MBA Entrance Exams
- CUET and UG Entrance
- Judiciary Exams
- Regulatory Body Exams
- Software Development