# Topological Notions (Wednesday, January 12)

:::{.remark}
Some topological notions to recall:

- $T_0$, Kolmogorov spaces: distinct points don't have the exact same neighborhoods, i.e. there exists a neighborhood of $x$ not containing $y$ **or** a neighborhood of $y$ not containing $x$.
- $T_1$, Frechet spaces: points are separated, so replace "or" with "and" above.
- $T_2$, Hausdorff spaces: points are separated by disjoint neighborhoods.
- Alexandrov spaces: arbitrary intersections of opens are open.
- Metrizability
- Paracompactness
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:::{.remark}
Recall that a topology $\tau$ on $X$ satisfies

- $\emptyset, X\in \tau$
- $A,B\in \tau \implies A \intersect B \in \tau$
- $\Union_{j\in J} A_j \in \tau$ if $A_j\in \tau$ for all $j$.

Equivalently one can specify the closed sets and require closure under finite unions and arbitrary intersections.
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:::{.example title="of topologies"}
Running examples:

- Any subset $S \subseteq \RR^n$ is Hausdorff and paracompact.
- Order topologies on posets
- Zariski topologies on varieties over $k= \bar{k}$, e.g. $\mspec A$ for $A\in \Alg^\fg\slice k$ or affine schemes $\spec A$.
- The discrete/initial topology $\tau = 2^X$.
- The indiscrete topology $\tau = \ts{\emptyset, X}$.
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:::{.remark}
Recall the separation axioms:

- $T_0$: points can be topologically distinguished.
  Note that the indiscrete topology s not $T_0$ if $\size X\geq 2$.

- $T_1$: points can be separated by (not necessarily disjoint) neighborhoods.
  Equivalently, points are closed.

- $T_2$/Hausdorff: points can be separated by disjoint neighborhoods.
- $T_{3.5}$/Tychonoff:?
- $T_6$:?

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:::{.exercise title="?"}
Show that points are closed in $X$ iff $X$ is $T_1$.
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:::{.definition title="Paracompactness"}
A space $X$ is **paracompact** iff every open cover $\mcu \covers X$ admits a *locally* finite refinement $\mcv \covers X$, i.e. any $x\in X$ is in only finitely many $V_i$.
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:::{.exercise title="Euclidean space is paracompact"}
Show that any $S \subseteq \RR^n$ is paracompact, and indeed any metric space is paracompact.
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:::{.solution}
Let $\mcu \covers X \da \RR^d$ be an open cover and define a proposed locally open refinement in the following way:

- Write $\mcu \da \ts{U_\alpha \st \alpha\in A}$ for some index set.
- Use that $W_n \da \cl_{X}(\BB_n(\vector 0))$ is compact, and since $\mcu \covers W_n$ there is a finite subcover $\mcv_n \da \ts{U_{n, 1}, \cdots, U_{n, m}}\covers \cl_X(\BB^n(\vector 0))^c$.
- Show that $\mcv \da \ts{\mcv}_{n\in \ZZ_{\geq 0}}$ is an open refinement of $\mcu$.
  - Why: it is a subcollection, and every $x\in X$ is in a ball of radius $R\approx N\da \ceil{\norm{x}}$.
  So $x\in \BB_N(0)$, thus $x\in U_{N, k}$ for some $k$.
- Show that $\mcv$ is locally finite.
  - Why: each $\mcv_n$ misses the $\BB_{k<n}(0)$, so each $x\not\in \Union_{k\geq N} \mcv_n$ if $N$ is defined as above.
  So $x$ is in only finitely many $\mcv_n$.
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:::{.fact}
Paracompact spaces admit a POU -- for $\mcu \covers X$, a collection $A$ of function $f_\alpha: X\to \RR$ where for all $\alpha \in \supp f_\alpha = \cl\qty{\ts{f\neq 0}}$, for all $x\in X$, there exists a $V\ni x$ such that for only finitely many $\alpha, \ro{f_\alpha}{V} \not\equiv 0$, and $\sum_{\alpha\in A} f_{ \alpha}(x) = 1$.
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:::{.remark}
Recall the order topology: for $(P, \leq )$ a poset, so

- $x\leq y, y\leq x\implies x=y$,
- $x\leq y\leq z\implies x\leq z$
- $x\leq x$

Define 

- Open sets to be increasing sets, so $x\in U, x\leq y \implies y\in U$, 
- Closed sets to be decreasing sets, so $x\in U, x\geq y \implies y\in U$

Note that this is a free choice!
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:::{.exercise title="?"}
Show that the order topology is closed under arbitrary unions *and* intersections of opens.
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:::{.exercise title="?"}
Show that the order topology is not $T_1$ by showing $\cl_P\qty{\ts{x}} = Z_{\leq}(x) \da \ts{y\in P\st y\leq x}$.
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:::{.fact}
For $k$ an infinite field, $\AA^1\slice k$ is the cofinite topology and thus not Hausdorff.
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