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The topology of the Sorgenfrey line $\mathbb{R}_{l}$ is definable in terms of its neighborhood systems $\bigl(\mathcal{N}_{x}\bigr)_{x \in \mathbb{R}}$, namely: $V \in \mathcal{N}_{x}$ if and only if $V$ contains some interval of the form $[x, y)$ with $y > x$. [Equivalently, the topology is definable in terms of the system $\bigl(\mathcal{B}_{x}\bigr)_{x \in \mathbb{R}}$ of local bases given by: $\mathcal{B}_{x} = \bigl\{[x, y) \, : y > x\bigr\}$.]

But is much simpler to define that topology as the one on the st $\mathbb{R}$ that has as base the collection of all half-open intervals $[a, b)$.

On the other hand, the "line with two origins" is probably most simply definable in terms of its neighborhood systems: Let $X = \mathbb{R} \cup \{z\}$, where $z \notin \mathbb{R}$. For $x \in \mathbb{R}$, take $\mathcal{N}_{x}$ to be the collection of all subsets of $X$ that contain a usual neighborhood of $x$; and take $\mathcal{N}_{z}$ to be the collection of all subsets of $X$ than contain $V \cup \{z\} \setminus \{0\}$ for some usual neighborhood $V$ of $x$.

Question:

What are some other simple examples of constructing a topology from its neighborhood systems — or, equivalently, from local bases of its neighborhood systems — where doing so is simpler than giving a base (or subbase)?

By saying "simple examples" I mean to suggest that the example should not involve, e.g., groups or function spaces.

Note: Of course, such standard things as product topologies and quotient topologies are most simply and directly defined in terms of bases (or subbases).

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  • $\begingroup$ In what sense do you consider your definition to be simpler than defining the basis to be $\{ (a, b) : a < b \} \cup \{ (a, 0) \cup (0, b) \cup \{ z \} : a < 0 < b \}$? $\endgroup$ – Daniel Schepler Jul 6 '17 at 17:12
  • $\begingroup$ We can also define a topology by its closure operator. This is often convenient for metric spaces. For example, two metrics on the same set generate the same topology iff they have the same convergent sequences. $\endgroup$ – DanielWainfleet Jul 6 '17 at 18:25
  • $\begingroup$ @DanielSchepler: Touché re the line with two origins! $\endgroup$ – murray Jul 8 '17 at 16:38
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Let $p$ be a prime number. If I wanted to define the $p$-adic topology on $\mathbb Q$ and if, for some reason, I had to decide between defining it from a base or defining it from its neighborhood systems, I would choose the later option: for each $q\in\mathbb Q$ and each $n\in\mathbb N$, consider the set$$N_{q,n}=\left\{q+\frac{ap^n}b\,\middle|\,a\in\mathbb{Z}\wedge b\in\mathbb{N}\wedge p\nmid b\right\}.$$Then a subset of $\mathbb Q$ is a neighborhood of $q$ if and only if it contains some set of the type $N_{q,n}$.

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  • $\begingroup$ Is the $p$-adic topology on $\mathbb{Q}$ not more naturally defined in terms of the $p$-adic metric (equivalently, in terms of the $p$-adic norm)? $\endgroup$ – murray Jul 6 '17 at 15:36
  • $\begingroup$ @murray Sure. That's why I wrote “if, for some reason, I had to decide between defining it from a base or defining it from its neighborhood systems”. $\endgroup$ – José Carlos Santos Jul 6 '17 at 15:37
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The topology on an arbitrary metric space is most naturally defined in terms of neighborhood bases $\mathcal{N}_x := \{ B_\epsilon(x) : \epsilon \in \mathbb{R}^+ \}$.

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  • $\begingroup$ Sort of "six of one half-dozen of the other": equally simple, the topology induced by a metric $d$ has the set of all balls $B_{\varepsilon}(x)$ of all radii $\varepsilon > 0$ at all points $x$. $\endgroup$ – murray Jul 6 '17 at 17:03
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    $\begingroup$ You could say the same thing about any collection of neighborhood bases: $\bigcup_{x \in X} \mathcal{N}_x$ is a basis for the topological space. $\endgroup$ – Daniel Schepler Jul 6 '17 at 17:04
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the topology of $\kappa \mathbb{N}$ (the Katetov extension) can be defined by local bases:

$X =\mathbb{N} \cup \{\mathcal{F}: \text{ free ultrafilter on } \mathbb{N}\}$

where $n \in \mathbb{N}$ has local base $\{\{n\}\}$ and $\mathcal{F}$ has local base $\{\{\mathcal{F}\} \cup A : A \in \mathcal{F}\}\}$.

Another favourite : if $\mathcal{A}$ is an almost disjoint family of subsets of $\mathbb{N}$ (which means any two different $A, B \in \mathcal{A}$ have a finite intersection), then $\Psi(\mathcal{A})$ (Mrowka space based on $\mathcal{A}$) is the set $\mathcal{N} \cup \{x_A : A \in \mathcal{A}\}$ where a local base for $n \in \mathcal{N}$ is again $\{\{n\}\}$ and for $x_A$ ($A \in \mathcal{A}$) it's all sets $N_m(A) = \{x_A\} \cup \{n \in A: n > m\}, m \in \mathbb{N}$. These local bases make it clear that $\Psi(\mathcal{A})$ is locally compact, and first countable and separable (as $\mathbb{N}$ is dense), while $\{x_A: A \in \mathcal{A}\}$ is closed and discrete.

Also the Niemytszki or Moore plane is best described in terms of local bases as well.

The rational sequence topology is quite similar to Mrowka space, but a bit more elementary : For every irrational number $x$, fix a sequence of rationals $q_n^x$ that converges to $x$ in the reals. (e.g. the truncated decimals expansions or partial continued fractions). Then a topology on $\mathbb{R}$ defined by local bases:

$\{\{x\}\}$ for $x \in \mathbb{Q}$, and $N_m(x) = \{x\} \cup \{q^x_n: n > m\}$ for $ m \in \mathbb{N}$ for $x \in \mathbb{R}\setminus \mathbb{Q}$ .

A theme emerges: this works best if there are different kinds of points in $X$, that have different local behaviour.

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  • $\begingroup$ The comment at the end reminded me of a yet more elementary construction: suppose $y_0 \notin X$. Then $Y := X \sqcup \{ y_0 \}$ has topology defined by neighborhood systems where for $x \in X$, any subset of $Y$ containing $x$ is a neighborhood; whereas the neighborhoods of $y_0$ are the cofinite subsets of $Y$ containing $y_0$. $\endgroup$ – Daniel Schepler Jul 6 '17 at 23:33
  • $\begingroup$ @Henno Brandsma: Thanks for reminding me of the "tangent disk space" (aka, the Moore plane). The Katetov extension is hardly "simple", involving as it does ultrafilters! I'll have to mull over the other two examples you gave; neither seems all that simple. $\endgroup$ – murray Jul 7 '17 at 14:35
  • $\begingroup$ @murray You don't need to know a lot about ultrafilters to define the space. $\endgroup$ – Henno Brandsma Jul 7 '17 at 15:05
  • $\begingroup$ @Daniel Schepler: now that really is a simple example! Does it have a name? (It's a "twist" on what Steen & Seebach, in Counterexamples in Topology, call a "Fort space".) $\endgroup$ – murray Jul 7 '17 at 15:18
  • $\begingroup$ @DanielSchepler that is just the one-point compactification of the discrete topology on $X$. $\endgroup$ – Henno Brandsma Jul 7 '17 at 15:20

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