Closure, interior, and frontier of preimages under re and im

In this fact we use the fact that ℂ is naturally homeomorphic to ℝ × ℝ to deduce some topological properties of Complex.re and Complex.im.

Main statements

Each statement about Complex.re listed below has a counterpart about Complex.im.

• Complex.isHomeomorphicTrivialFiberBundle_re: Complex.re turns ℂ into a trivial topological fiber bundle over ℝ;
• Complex.isOpenMap_re, Complex.quotientMap_re: in particular, Complex.re is an open map and is a quotient map;
• Complex.interior_preimage_re, Complex.closure_preimage_re, Complex.frontier_preimage_re: formulas for interior (Complex.re ⁻¹' s) etc;
• Complex.interior_setOf_re_le etc: particular cases of the above formulas in the cases when s is one of the infinite intervals Set.Ioi a, Set.Ici a, Set.Iio a, and Set.Iic a, formulated as interior {z : ℂ | z.re ≤ a} = {z | z.re < a} etc.

Tags

complex, real part, imaginary part, closure, interior, frontier

theorem Complex.isHomeomorphicTrivialFiberBundle_re : IsHomeomorphicTrivialFiberBundle ℝ Complex.re

Complex.re turns ℂ into a trivial topological fiber bundle over ℝ.

theorem Complex.isHomeomorphicTrivialFiberBundle_im : IsHomeomorphicTrivialFiberBundle ℝ Complex.im

Complex.im turns ℂ into a trivial topological fiber bundle over ℝ.

theorem Complex.isOpenMap_re : IsOpenMap Complex.re

theorem Complex.isOpenMap_im : IsOpenMap Complex.im

theorem Complex.quotientMap_re : QuotientMap Complex.re

theorem Complex.quotientMap_im : QuotientMap Complex.im

theorem Complex.interior_preimage_re (s : Set ℝ) : interior (Complex.re ⁻¹' s) = Complex.re ⁻¹' interior s

theorem Complex.interior_preimage_im (s : Set ℝ) : interior (Complex.im ⁻¹' s) = Complex.im ⁻¹' interior s

theorem Complex.closure_preimage_re (s : Set ℝ) : closure (Complex.re ⁻¹' s) = Complex.re ⁻¹' closure s

theorem Complex.closure_preimage_im (s : Set ℝ) : closure (Complex.im ⁻¹' s) = Complex.im ⁻¹' closure s

theorem Complex.frontier_preimage_re (s : Set ℝ) : frontier (Complex.re ⁻¹' s) = Complex.re ⁻¹' frontier s

theorem Complex.frontier_preimage_im (s : Set ℝ) : frontier (Complex.im ⁻¹' s) = Complex.im ⁻¹' frontier s

@[simp]

theorem Complex.interior_setOf_re_le (a : ℝ) : interior {z : ℂ | z.re ≤ a} = {z : ℂ | z.re < a}

@[simp]

theorem Complex.interior_setOf_im_le (a : ℝ) : interior {z : ℂ | z.im ≤ a} = {z : ℂ | z.im < a}

@[simp]

theorem Complex.interior_setOf_le_re (a : ℝ) : interior {z : ℂ | a ≤ z.re} = {z : ℂ | a < z.re}

@[simp]

theorem Complex.interior_setOf_le_im (a : ℝ) : interior {z : ℂ | a ≤ z.im} = {z : ℂ | a < z.im}

@[simp]

theorem Complex.closure_setOf_re_lt (a : ℝ) : closure {z : ℂ | z.re < a} = {z : ℂ | z.re ≤ a}

@[simp]

theorem Complex.closure_setOf_im_lt (a : ℝ) : closure {z : ℂ | z.im < a} = {z : ℂ | z.im ≤ a}

@[simp]

theorem Complex.closure_setOf_lt_re (a : ℝ) : closure {z : ℂ | a < z.re} = {z : ℂ | a ≤ z.re}

@[simp]

theorem Complex.closure_setOf_lt_im (a : ℝ) : closure {z : ℂ | a < z.im} = {z : ℂ | a ≤ z.im}

@[simp]

theorem Complex.frontier_setOf_re_le (a : ℝ) : frontier {z : ℂ | z.re ≤ a} = {z : ℂ | z.re = a}

@[simp]

theorem Complex.frontier_setOf_im_le (a : ℝ) : frontier {z : ℂ | z.im ≤ a} = {z : ℂ | z.im = a}

@[simp]

theorem Complex.frontier_setOf_le_re (a : ℝ) : frontier {z : ℂ | a ≤ z.re} = {z : ℂ | z.re = a}

@[simp]

theorem Complex.frontier_setOf_le_im (a : ℝ) : frontier {z : ℂ | a ≤ z.im} = {z : ℂ | z.im = a}

@[simp]

theorem Complex.frontier_setOf_re_lt (a : ℝ) : frontier {z : ℂ | z.re < a} = {z : ℂ | z.re = a}

@[simp]

theorem Complex.frontier_setOf_im_lt (a : ℝ) : frontier {z : ℂ | z.im < a} = {z : ℂ | z.im = a}

@[simp]

theorem Complex.frontier_setOf_lt_re (a : ℝ) : frontier {z : ℂ | a < z.re} = {z : ℂ | z.re = a}

@[simp]

theorem Complex.frontier_setOf_lt_im (a : ℝ) : frontier {z : ℂ | a < z.im} = {z : ℂ | z.im = a}

theorem Complex.closure_reProdIm (s : Set ℝ) (t : Set ℝ) : closure (s ×ℂ t) = closure s ×ℂ closure t

theorem Complex.interior_reProdIm (s : Set ℝ) (t : Set ℝ) : interior (s ×ℂ t) = interior s ×ℂ interior t

theorem Complex.frontier_reProdIm (s : Set ℝ) (t : Set ℝ) : frontier (s ×ℂ t) = closure s ×ℂ frontier t ∪ frontier s ×ℂ closure t

theorem Complex.frontier_setOf_le_re_and_le_im (a : ℝ) (b : ℝ) : frontier {z : ℂ | a ≤ z.re ∧ b ≤ z.im} = {z : ℂ | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ b ≤ z.im}

theorem Complex.frontier_setOf_le_re_and_im_le (a : ℝ) (b : ℝ) : frontier {z : ℂ | a ≤ z.re ∧ z.im ≤ b} = {z : ℂ | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ z.im ≤ b}

theorem IsOpen.reProdIm {s : Set ℝ} {t : Set ℝ} (hs : IsOpen s) (ht : IsOpen t) : IsOpen (s ×ℂ t)

theorem IsClosed.reProdIm {s : Set ℝ} {t : Set ℝ} (hs : IsClosed s) (ht : IsClosed t) : IsClosed (s ×ℂ t)

theorem Bornology.IsBounded.reProdIm {s : Set ℝ} {t : Set ℝ} (hs : Bornology.IsBounded s) (ht : Bornology.IsBounded t) : Bornology.IsBounded (s ×ℂ t)