Metric Spaces

(b) fails for $A=\mathbb{Q},B=\mathbb{R}$∖$\mathbb{Q}.\partial A=$ cl A ∩ cl(Aᶜ) is an intersection of closed sets.

Every open interval contains irrationals, so no open interval fits inside $\mathbb{Q}$ — the interior is empty. (Symmetrically, the interior of the irrationals is also empty, while both sets are dense.)

(1) A limit of limit points of A is itself approached by points of A, so A′ contains its own limit points. (2) Standard characterisation of closure. (3) Around any point one can shrink a ball to exclude the finitely many other members of A. (4) Fails: A = {1/n : n ≥ 1} has 0 as a limit point, and 0 ∉ A.

(a) True — closed and bounded in $\mathbb{R}.(b)$ True — compact ⇒ complete. (c) False — 0 is a limit point; f(0)=1, f(1/n)=0 is discontinuous. (d) False — {1} is both open and closed in X.

By Heine–Borel a subset of $\mathbb{R}$ is compact exactly when it is closed and bounded. (1) Bounded, and closed because the only limit point 0 has been included. (2) Bounded but not closed — an irrational in (0,1) is a limit point that is missing. (3) The Cantor set is a bounded intersection of closed sets, hence closed. (4) Bounded but not closed: 0 is a limit point and is absent — this is exactly (1) with the limit point removed.

Heine–Borel: compact in ${\mathbb{R}}^2=$ closed and bounded. The closed disc is both. The open disc is not closed, the first quadrant is unbounded, and the hyperbola is closed but unbounded.

$T^{\prime }(x)=1-1/x^2\in (0,1)$ so distances strictly shrink, but no uniform k < 1 exists and x + 1/x = x is impossible.

(d) is true (singletons) but trivially, not via Baire; the other three are standard Baire consequences.

$\mathbb{N}$ is closed in $\mathbb{R}($and discrete$).\mathbb{Q}$ and (0,1] have escaping Cauchy sequences; the $L^1$ metric on continuous functions is incomplete.

$(1)\mathbb{R}$ is a complete metric space, and Baire says such spaces are of second category in themselves. $(2)\mathbb{Q}=\cup${q} and each singleton is nowhere dense in $\mathbb{R}.(3)$ Fails: if the irrationals were first category, adding the first-category set $\mathbb{Q}$ would make $\mathbb{R}$ first category, contradicting (1). (4) With no isolated points each singleton is nowhere dense; a countable such space would be a countable union of nowhere dense singletons, contradicting Baire.

X is the closure of the connected graph, hence connected; no path from the segment reaches the graph because sin(1/x) oscillates.

(a) det < 0 and det > 0 are disjoint open pieces. (b) SL${}_2(\mathbb{R})$ is path-connected. (c) the complement of a countable set in ${\mathbb{R}}^2$ is path-connected: through any two points pass uncountably many lines, avoid the countably many bad ones. (d) O(2) splits by det = ±1.

(1) The hyperbola xy = 1 has two branches in opposite quadrants; the open half-planes x > 0 and x < 0 disconnect it. (2) The circle is a continuous image of $[0,2\pi ].(3)$ The union of the two axes is connected: each axis is connected and they share the origin. (4) The punctured plane is path-connected — any two points join by a path avoiding one point — unlike the punctured line $\mathbb{R}$ \ {0}.