Short Answer 
Only Option B (‚àí‚àû, ‚àí1) is considered an open set because it does not include its boundary point, while the other options contain boundary points or fail to meet the criteria for being open. Understanding the definition of open sets is crucial in real analysis.
Step 1: Understand Open Sets
To determine if a set is open, we must examine its boundary points. A set is considered open in the real line if none of its boundary points are included within it. This means that for any point in the set, we should be able to find a small interval around that point that also lies completely within the set.
Step 2: Analyze Each Option
Now, we will evaluate each option to see if it is open or not. Here are the evaluations:
- Option A: A ‚à© [0, 1] results in {0}, which includes boundary point 0, so it’s not open.
- Option B: A ‚à© (‚àí‚àû, ‚àí1) results in (‚àí‚àû, ‚àí1), which doesn’t include the boundary point ‚àí1, making it open.
- Option C: A ‚à™ {1/2} results in (‚àí‚àû, 0] ‚à™ {1/2}, includes boundary point 0, so not open.
- Option D: A ‚àñ (‚àí1, 1) results in (‚àí‚àû, ‚àí1] ‚à™ {0}, includes both ‚àí1 and 0, so not open.
- Option E: A ‚à™ (0.1, 1) results in (‚àí‚àû, 0] ‚à™ (0.1, 1), still includes boundary point 0, so not open.
Step 3: Conclusion
After analyzing each option, we conclude that only Option B (‚àí‚àû, ‚àí1) is an open set. The other options either contain their boundary points or do not meet the criteria for being an open set. Understanding these properties is key to distinguishing open sets in real analysis.