- #1
yifli
- 68
- 0
Is my claim correct?
Boundary of any set in a topological space is compact
- Context: Graduate
- Thread starter yifli
- Start date
-
- Tags
- Boundary Compact Set Space Topological
Click For Summary
SUMMARY
The claim that the boundary of any set in a topological space is compact is incorrect, as demonstrated by the set of rational numbers, \mathbb{Q}, whose boundary is the entire real line, \mathbb{R}, which is not compact. However, this result holds true in compact topological spaces, where any closed set, including boundaries, is compact. This distinction is crucial for understanding the properties of boundaries in different types of topological spaces.
PREREQUISITES- Understanding of basic topology concepts, including open and closed sets.
- Familiarity with compact spaces and their properties.
- Knowledge of real numbers and the set of rational numbers, \mathbb{Q}.
- Basic mathematical notation and terminology used in topology.
- Study the properties of compact topological spaces in detail.
- Learn about closed sets and their significance in topology.
- Explore examples of boundaries in various topological spaces.
- Investigate the implications of compactness in different mathematical contexts.
Mathematicians, students of topology, and anyone interested in the properties of sets and boundaries in topological spaces.
- #2
- 22,170
- 3,333
Hi yifli! 
No, this is not correct, not even in the nice space [itex]\mathbb{R}[/itex]. Indeed, the set of rationals [itex]\mathbb{Q}[/itex] has a boundary which is entire [itex]\mathbb{R}[/itex] and is thus not compact!
The result is true in compact topological spaces, however. (because any closed set in a compact space is compact, and because the boundary is always a closed set).
No, this is not correct, not even in the nice space [itex]\mathbb{R}[/itex]. Indeed, the set of rationals [itex]\mathbb{Q}[/itex] has a boundary which is entire [itex]\mathbb{R}[/itex] and is thus not compact!
The result is true in compact topological spaces, however. (because any closed set in a compact space is compact, and because the boundary is always a closed set).
Similar threads
- · Replies 2 ·
High School
Sequential Compactness of Sets: A Definition
- · Replies 10 ·
Undergrad
On two definitions of locally compact
- · Replies 5 ·
- · Replies 5 ·
- · Replies 11 ·
- · Replies 3 ·
- · Replies 12 ·
Undergrad
Preservation of Local Compactness
- · Replies 4 ·
- · Replies 14 ·
- · Replies 2 ·