Summary of "Lecture 10(B): Euclidean Space: Examples and Theorems"
Main ideas, concepts, and lessons
- The lecture is about building intuition for open sets, neighborhoods, ε-balls, and closed sets in Euclidean space (starting in (\mathbb{R}), then (\mathbb{R}^2), and (\mathbb{R}^N)).
- Key recurring pattern:
- Sets defined by strict inequalities (e.g., (x<0), (x>1), (U(x)>C), (P\cdot x>0)) are treated as open (often because points can fit inside an ε-ball staying within the set).
- Sets defined by non-strict equalities (e.g., (x=0) boundary, (U(x)=C) level curve, (P\cdot x=0) hyperplane) are closed as complements of open sets.
Methodology / instruction-like steps shown
How to show a set is open (ε-ball criterion)
To prove a set (S) is open:
- Take an arbitrary point (x \in S).
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Show that there exists an (\varepsilon>0) such that the open ball [ B(x,\varepsilon)={y:|y-x|<\varepsilon} ] satisfies: [ B(x,\varepsilon)\subseteq S. ]
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Repeat for every point in (S).
Theorem proof strategy: union of two open sets is open
Goal: Prove if (S_1) and (S_2) are open, then (S_1\cup S_2) is open.
- Start with an arbitrary point (x\in S_1\cup S_2).
- Then (x\in S_1) or (x\in S_2) (possibly both).
Case 1: If (x\in S_1)
- Since (S_1) is open, there exists (\varepsilon>0) such that (B(x,\varepsilon)\subseteq S_1).
- Conclude (B(x,\varepsilon)\subseteq S_1\cup S_2) (because (S_1\subseteq S_1\cup S_2)).
Case 2: If (x\in S_2)
- Similarly, there exists (\varepsilon>0) such that (B(x,\varepsilon)\subseteq S_2).
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Conclude (B(x,\varepsilon)\subseteq S_1\cup S_2).
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Therefore, every point in (S_1\cup S_2) has an ε-ball contained in the union, so (S_1\cup S_2) is open.
Closed set construction via complements
- Use the fact:
- A set is closed iff its complement is open.
- Example used:
- If an open set is (S), then its complement (S^c) is closed.
Intersection of open sets is open (stated as the next theorem/exercise)
- If (S_1) and (S_2) are open, then (S_1\cap S_2) is open.
- This is described as an “easy exercise,” with the claim that it follows directly from the open-set definition.
Vacuous truth for the empty set
- The empty set (\varnothing) is considered open.
- Reason (as explained):
- The definition of open sets requires a condition for every point in the set.
- Since (\varnothing) has no points, the condition holds automatically (a “proof by vacuity”).
Deduction: (\mathbb{R}^N) is both open and closed
- Since (\varnothing) is open, its complement (\mathbb{R}^N) is closed.
- Also, (\mathbb{R}^N) is open (as part of the general reasoning that complements preserve open/closed status).
- So (\mathbb{R}^N) and (\varnothing) are both open and closed.
Examples covered (with main point of each)
Example (in (\mathbb{R})): two disjoint open rays and their union
- Sets:
- ((-\infty,0)) and ((1,\infty)) are treated as open in (\mathbb{R}).
- Main point:
- For a point (x<0) in ((-\infty,0)), choose an ε-ball small enough to stay left of 0.
- For a point (x>1) in ((1,\infty)), choose ε small enough to stay right of 1.
- The union of these two open sets is open (motivating the union theorem).
Complement gives a closed set
- The complement of ((-\infty,0)\cup(1,\infty)) is the closed interval ([0,1]).
- Lecture emphasis:
- If you already know the union is open, its complement is automatically closed.
Example: half-open/half-closed interval is neither open nor closed
- The interval ((0,1]):
- Not open because points near (1) require neighborhoods that include points (>1), which are not in the set.
- Not closed because its complement is not open (the complement includes (0), causing failure of openness).
Example: utility/upper contour sets in economics geometry ((\mathbb{R}^2))
- A “utility function” (U(x_1,x_2)) with level curves (U(x)=C).
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Defined sets:
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Indifference/level curve: [ S={x\in \mathbb{R}_+^2: U(x)=C} ]
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Upper set (upper contour): [ U={x\in \mathbb{R}_+^2: U(x)>C} ]
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Lower set: [ L={x\in \mathbb{R}_+^2: U(x)<C} ]
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Claim (stated, with continuity of (U) assumed): if (U) is continuous, then:
- The upper set (U(x)>C) is open.
- The lower set (U(x)<C) is open.
- Then the complement of ((\text{upper set})\cup(\text{lower set})) is the level curve, so the level curve is closed.
- Connection to microeconomics:
- This open-set structure will recur when analyzing indifference curves and feasible changes.
Example: hyperplanes/half-spaces defined by a dot product ((\mathbb{R}^N))
Let (P\neq 0) in (\mathbb{R}^N). Define:
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Hyperplane: [ H={x\in\mathbb{R}^N: P\cdot x=0} ]
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Half-spaces: [ H^+={x: P\cdot x>0},\quad H^-={x: P\cdot x<0} ]
Claims (used without full analytic proof in the lecture):
- (H^+) is open.
- (H^-) is open.
- Therefore (H^+\cup H^-) is open.
- The complement is exactly the hyperplane (H), so (H) is closed.
Economics interpretation:
- (P\cdot x) is interpreted like a “value” of an activity vector (x) given prices (P).
Speaker / sources featured
- Speaker: Not explicitly named in the provided subtitles (lecture-style instruction from an unnamed instructor).
- Sources/citations: None explicitly referenced.
Category
Educational
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