5 questions
Mathematics
Algebra, calculus, analysis, topology, linear algebra and number theory, worked out step by step.
real analysis 3general topology 3open sets 3metric spaces 2connectedness 1graph theory 1euler path 1puzzles 1combinatorics 1calculus 1limits 1asymptotes 1rational functions 1
Calculators
Matrix
Determinant with cofactor expansion, inverse, symmetric eigen-decomposition and singular values.
Volume
Volume of a box or cylinder in cubic metres, litres, gallons and cubic feet from any length unit.
Questions
- Are Singleton Sets in ℝ Both Open and Closed?In ℝ with the usual topology, a singleton {a} is closed but not open. Proofs of both facts, and why only ∅ and ℝ are both open and closed.
- Can You Draw This Figure Without Lifting the Pen? (Euler Paths)A figure can be drawn in one stroke exactly when it is connected and has 0 or 2 points where an odd number of lines meet. How to check, and where to start.
- Is Every Singleton Set Open? (No, Here's When It Is)Singleton sets are not open in general. In ℝ, {0} is closed but not open. A singleton {x} is open exactly when x is an isolated point, as in discrete spaces.
- How to Find Horizontal and Vertical Asymptotes Using LimitsHorizontal asymptotes come from limits as x → ±∞; vertical asymptotes from infinite one-sided limits at a point. Step-by-step method with worked examples.
- Why Is Every Singleton Set Open in a Finite Metric Space?Proof that {x} is open in any finite metric space, using the minimum distance to the other points, and why this makes every subset open (the discrete topology).