The topologist's sine curve is connected but not path connected

Given: The set $S$ in the Statement and the two points $(0,0),(1,\sin 1)\in S$.

The topologist's sine curve $S=\{(x,\sin (1\mathrm{/}x)):0<x\le 1\}\cup (\{0\}\times [-1,1])$ is connected (The topologist's sine curve is connected).

Projections from a product are continuous (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice, claim 1).

For a continuous real function on a subset of $\mathbb{R}$, the preimage of a closed set is relatively closed ($f:A\to \mathbb{R}$ is continuous on $A$ if and only if the preimage of every open subset of $\mathbb{R}$ is the intersection with $A$ of an open subset of $\mathbb{R}$, and dually for closed sets); real and metric continuity agree for the usual metric (Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, claim 1).

The interval $[0,1]$ is compact (Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact), real and metric compactness agree on subsets of $\mathbb{R}$ (Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace, claim 5), and a closed subset of a compact metric space is compact (A closed subset of a compact metric space is compact).

A continuous real function on a nonempty compact metric space attains its maximum (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value).

A continuous real function on a connected space attains every intermediate value between two of its values (A real-valued continuous map on a connected space has order-convex image, so it takes every value between any two of its values).

The quarter-turn values and period give $\sin (\pi \mathrm{/}2+2m\pi )=1$ and $\sin (3\pi \mathrm{/}2+2m\pi )=-1$ for every integer $m$, and for every real $\epsilon >0$ some positive integer $N$ satisfies $1\mathrm{/}N<\epsilon$ (Quarter-turn values and shifts by pi/2 and pi, The zero sets of sine and cosine and the least positive common period 2 pi, For every $\epsilon >0$ in a complete ordered field there is a natural $n\ge 1$ with $1\mathrm{/}n<\epsilon$).

A space is path connected when every pair of points is joined by a continuous path from $[0,1]$ (Paths, path-connected spaces and path components).

The number $\pi =2\gamma$ is positive because the smallest positive zero of cosine satisfies $\gamma \in (0,2)$ (Pi as twice the smallest positive zero of cosine, Cosine has a smallest positive zero, lying strictly between zero and two).

Every interval in the real line, including $[s,r]$, is connected (The connected subspaces of $\mathbb{R}$ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in $\mathbb{R}$").

A composite of continuous maps is continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous, claim 1).