What is topology? — Mathematics

Definition 25.1 University Mathematics — Year 1 · Chapter 25 — Functions of Two Variables

On $\R^2$ , use the Euclidean norm $\norm{(x,y)} = \sqrt{x^2 + y^2}$ (Chapter 23). Open balls, neighborhoods, open subsets of $\R^2$ are defined exactly as in Chapter 12, with balls in place of intervals. A function $f \colon U \to \R$ ( $U \subseteq \R^2$ open) is continuous at $a \in U$ when

\forall\varepsilon > 0,\ \exists\delta > 0, \quad \norm{X - a} \leq \delta \implies \abs{f(X) - f(a)} \leq \varepsilon,

with the same sequential characterization as in one variable. Sums, products, quotients and compositions with continuous one-variable functions preserve continuity; the coordinate maps are continuous, hence so are polynomials in $(x,y)$ .

Examples

Example 25.2 (The polar bound, the clean way to prove a limit)

Show that $f(x, y) = \dfrac{x^2y^2}{x^2 + y^2}$ (with $f(0,0) = 0$ ) is continuous at the origin. In polar coordinates $x = \rho\cos\theta$ , $y = \rho\sin\theta$ :

\abs{f} = \frac{\rho^4\cos^2\theta\sin^2\theta}{\rho^2} = \rho^2\,(\cos\theta\sin\theta)^2 \leq \frac{\rho^2}{4} \xrightarrow[\rho \to 0]{} 0 ,

a bound independent of $\theta$ : whatever the direction of approach, the values are squeezed to $0$ . That uniformity in $\theta$ is the whole point — a bound like $\abs g = \abs{\cos\theta\sin\theta}$ (no $\rho$ left) proves nothing, and indeed that $g$ is the discontinuous radial trap of the next example.

Example 25.3 (The radial trap)

Let $f(x, y) = \dfrac{xy}{x^2 + y^2}$ for $(x,y) \neq (0,0)$ , $f(0, 0) = 0$ . Along each axis, $f = 0 \to 0$ ; but along the diagonal $y = x$ , $f(x, x) = \frac12 \not\to 0$ . No limit at the origin: approaching along every line, and even finding the same limit along each, is not enough (here the line limits disagree; worse examples agree along all lines yet fail along a parabola, Exercise 25.3). Continuity in each variable separately does not imply continuity.

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Definition 6.1 University Mathematics — Year 3 · Chapter 6 — General Topology

A topology on a set $X$ is a family $\mathcal T$ of subsets of $X$ — called open sets — such that: $\varnothing, X \in \mathcal T$ ; any union of open sets is open; any finite intersection of open sets is open. The pair $(X, \mathcal T)$ is a topological space. Complements of open sets are closed. A neighborhood of $x$ is a set containing an open set containing $x$ .

Examples

Example 6.2

(a) A metric space, with “open” as in Year 2 (unions of open balls): the metric topology; different metrics can give the same topology (equivalent metrics). A space whose topology arises from some metric is metrizable. (b) The discrete topology (all subsets) and the indiscrete topology $\{\varnothing, X\}$ . (c) The cofinite topology on an infinite set: open $=$ empty or cofinite. Not metrizable, as we shall see (Exercise 6.3). (d) On $\R$ , the usual topology; on $\bar\R = \R \cup \{\pm\infty\}$ , the order topology generated by rays — making “ $x_n \to +\infty$ ” an instance of plain convergence.

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Examples

Examples

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