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The quotient map is open, and every interval shorter than one embeds in $R / Z$

Statement

Let $p : R \to R / Z$ be the quotient map of The circle as $S^{1} = R / Z$ with basepoint $[0]$ . The quotient map is open, and every interval shorter than one embeds in $R / Z$ .

More precisely, for every open $U \subseteq R$ ,

$p^{- 1} (p [U]) = ⋃_{n \in Z} (U + n),$

and $p [U]$ is open. If $a \leq b$ , $b - a < 1$ , and $J$ is any of $(a, b)$ , $[a, b]$ , $[a, b)$ , or $(a, b]$ , then $p ∣_{J} : J \to p [J]$ is a homeomorphism, with both sides carrying their subspace topologies.

Facts & Assumptions

Given: The quotient projection $p$ , an open set $U \subseteq R$ , and an interval $J$ of one of the displayed four forms with length $ℓ = b - a < 1$ .

[L1]

Let $p : R \to R / Z$ be the quotient projection inducing the quotient topology, with $p (x) = [x]$ and $p (x) = p (y)$ exactly when $x - y \in Z$ (The circle as $S^{1} = R / Z$ with basepoint $[0]$ ).

[L2]

Identify $Z$ with its canonical copy inside $R$ . Then for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ (Integer part: for every real $x$ there is exactly one integer $m$ with $m \leq x < m + 1$ ).

[L3]

A function $f : X \to Y$ is an open map if $f [V]$ is open in $Y$ for every open $V \subseteq X$ ; an embedding is a homeomorphism onto its image with the subspace topology (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological).

[L4]

Every constant real-valued function and the identity are continuous, and finite sums and scalar multiples of continuous real-valued functions are continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function).

Proof

technique · direct

1.1

A real $x$ lies in $p^{- 1} (p [U])$ exactly when $p (x) = p (u)$ for some $u \in U$ , which by [L1] is equivalent to $x - u = n$ for some $n \in Z$ ; hence $p^{- 1} (p [U]) = ⋃_{n \in Z} (U + n)$ . Each translate $U + n$ is open: translating by $n$ and by $- n$ gives mutually inverse continuous maps by [L4]. The union is open, so the quotient-topology criterion in [L1] makes $p [U]$ open. Thus $p$ is open in the sense of [L3], including when $U = \emptyset$ .

L1L3L4

1.2

Suppose $x, y \in J$ and $p (x) = p (y)$ . Then $k : = x - y \in Z$ by [L1], while $∣ k ∣ = ∣ x - y ∣ \leq ℓ < 1$ . If $0 < k < 1$ , both $0$ and $k$ satisfy the integer-part inequalities for the real $k$ , contrary to uniqueness in [L2]; applying the same argument to $- k$ excludes $- 1 < k < 0$ . Hence $k = 0$ and $x = y$ , so $p ∣_{J}$ is injective. This also covers a singleton interval; for an empty interval injectivity is vacuous.

L1L2algebra

2.1

The restriction $p ∣_{J}$ is continuous and is a bijection onto $p [J]$ by step 1.2. To prove its inverse continuous, let $O$ be relatively open in $J$ and $x \in O$ . Choose $δ > 0$ with $J \cap (x - δ, x + δ) \subseteq O$ , and put $r = \frac{1}{2} \min {δ, 1 - ℓ} > 0$ . Step 1.1 makes $W : = p [(x - r, x + r)]$ open. If $z \in J$ and $p (z) \in W$ , choose $y \in (x - r, x + r)$ with $p (z) = p (y)$ ; then $z - y \in Z$ by [L1] and $∣ z - y ∣ \leq ∣ z - x ∣ + ∣ x - y ∣ < ℓ + r < 1$ , so [L2] gives $z = y$ . Thus $z \in J \cap (x - δ, x + δ) \subseteq O$ , and $W \cap p [J] \subseteq p [O]$ . Every point of $p [O]$ therefore has a relative open neighbourhood contained in $p [O]$ , so $p [O]$ is open in $p [J]$ . The empty case has the unique empty inverse. Hence $p ∣_{J}$ is a homeomorphism onto its image, and therefore an embedding by [L3].

step 1.1step 1.2L1L2L3L4algebra∎

Depends on

Used by

An interval of length one need not embed under p:ℝ→ℝ/ℤ Counterexample
ℝ/ℤ is Hausdorff Proposition
p:ℝ→ℝ/ℤ is a covering map with translated interval sheets Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 102 results over 26 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Allen Hatcher, Algebraic Topology, Ch. 1, Section 1.1 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Ch. 1, Section 5 (standard reference, not scraped)
Jonathan Wise, Math 6210 Lecture Notes, Week 3, Sections 3.1 and 3.4 (standard reference, not scraped)

The quotient map is open, and every interval shorter than one embeds in R/ZR/Z