Recently, I started reading the book “Topology Without Tears” by Sidney Morris. It is such an exciting adventure. And as always, I write notes about what I learned.
Topology
Definition. Let X be a non-empty set and \(\mathcal{T}\) be a set of subsets of X, i.e. \(\mathcal{T}\subseteq\mathcal{P}(X)\). We call \(\mathcal{T}\) a topology on X if
- \(X,\emptyset\in\mathcal{T}\)
- The union of any (finite or infinite, countable or uncountable) number of sets that are in \(\mathcal{T}\) also belongs to \(\mathcal{T}\)
- The intersection of any two sets that are in \(\mathcal{T}\) also belongs to \(\mathcal{T}\).
The pair \((X, \mathcal{T})\) is called a topological space.
When \(\mathcal{T} = \mathcal{P}(X)\), it is called the discrete topology on X. Together they form the discrete topological space.
When \(\mathcal{T} = \{\emptyset, X\}\), it is called the indiscrete topology on X. Together they form the indiscrete topological space.
Notice the article “the (in)discrete topo”, it means for a non-empty set X, there is exactly ONE such topo.
Theorem. If \((X,\mathcal{T})\) is a topological space (from now abbreviated as t.s.) such that for every element \(x\in X\), the singleton set \(\{x\}\in\mathcal{T}\), then \(\mathcal{T}\) is a discrete topology (from now abbreviated as topo).
Open sets, closed sets, clopen sets
If \((X,\mathcal{T})\) is a t.s., each element of \(\mathcal{T}\) is called an open set. Meanwhile, we call a set \(S\subseteq X\) a closed set if \(X-S\in\mathcal{T}\), i.e. X-S is open.
Theorem. If \((X,\mathcal{T})\) is a t.s.,
- \(\emptyset, X\) are closed sets
- The intersection of any (finite or infinite) number of closed sets is a closed set
- The union of any two closed sets is a closed set
If S is both an open and a closed set, we call it a clopen set.
- In a discrete t.s., any subset of X is clopen.
- In an indiscrete t.s., only \(\emptyset\) and X are clopen.
- In any t.s., \(\emptyset\) and X are clopen.
The Finite-closed Topology
Definition. Let X be any non-empty set. A topo \(\mathcal{T}\) on X is called the finite-closed topology, or cofinite topology on X if the closed subsets of X consist of X and all finite subsets of X.
which also means that the open sets are \(\emptyset\) and subsets of X that have finite complements.
“consist of” is different from “include”. When way say “the closed subsets of X consist of X and all finite subsets of X”, it means no sets other than the specified sets are closed, only X and all finite subsets of X are closed.
Infinite sets are not necessarily open sets. If the complement of an infinite set is also an infinite set, it is definitely not an open set.
Let \(\mathcal{T}\) be the cofinite topo on X. If X has at least 3 distinct clopen subsets, X is finite. ▶ Proof
We know that X and \(\emptyset\) are always clopen. Let S be another clopen set. Since it is open, X-S is finite. Since it is closed, S is finite. Hence \(X=S\cup (X-S)\) is finite.
Some other topologies
\(T_1\)-space (Fréchet space)
A topo space \((X,\mathcal{T})\) is called a \(T_1\)-space if every singleton set \(\{x\}\) is closed. Another definition is that for every pair of distinct points a and b in X, there exists an open set containing a but not b, and there exists an open set containing b but not a. Examples:
- Any discrete space is a \(T_1\)-space, since \(\mathcal{T}=\{S:S\subset X\}\) and \(X-\{x\}\subset X\), which brings us to \(X-\{x\}\in\mathcal{T}\) (open set), so \(\{x\}\) is closed, \(\forall x\).
- An infinite set with the finite-closed topo is a \(T_1\)-space, because every singleton set is finite, and in the finite-closed topo space every finite set is closed.
\(T_0\)-space (Kolmogorov space)
A topo space \((X,\mathcal{T})\) is called a \(T_0\)-space if for each pair of distinct points a and b in X, there exists an open set containing a but not b, or there exists an open set containing b but not a. Examples:
- Any \(T_1\)-space is an \(T_0\)-space. For each pair of distinct points a and b, \(\{a\}\) is open and does not contain b.
- With the same explanation, any discrete space is a \(T_0\)-space.
Sierpiński space
It is a \(T_1\)-space on \(X=\{0,1\}\) but not a \(T_0\)-space, i.e. \(\mathcal{T}=\{\emptyset, \{0,1\}, \{0\}\}\) or \(\mathcal{T}=\{\emptyset, \{0,1\}, \{1\}\}\)
Countable-closed space
Let X be an infinite set. The countable-closed topology is defined to have X and all countable subsets of X as its closed sets.
Door space
A topo space \((X,\mathcal{T})\) is called a door space if every subset of X is open or closed.
Some topologies on \(\mathbb{R}\)
- The discrete topology
- The indiscrete topology
- The finite-closed topology
- \(\mathcal{T}\) consists of \(\mathbb{R}, \emptyset\), and every interval (-n, n) for n positive integer. ▶ Why?
We can safely say that \(\mathcal{T}\) is countable. Let’s prove that any union of elements in \(\mathcal{T}\) is also in \(\mathcal{T}\).
\[S = \bigcup_{n\in I} (-n, n),\ \mathrm{where}\ I\subseteq \mathbb{N}^+\]If I is finite, \(S=(-m,m)\) where m is the maximum element in I, hence it is also in \(\mathcal{T}\). Otherwise, I is infinite and equal to \(\mathbb{N}^+\), hence \(S=\mathbb{R}\in\mathcal{T}\).
It is easy to prove the intersection between two sets in \(\mathcal{T}\) is also in \(\mathcal{T}\).
- \(\mathcal{T}\) consists of \(\mathbb{R}, \emptyset\), and every interval [-n, n] for n positive integer.
- \(\mathcal{T}\) consists of \(\mathbb{R}, \emptyset\), and every interval \([n,\infty)\) for n positive integer.
- \(\mathcal{T}\) consists of \(\mathbb{R}, \emptyset\), and every interval (-r, r) for r positive real.
- \(\mathcal{T}\) consists of \(\mathbb{R}, \emptyset\), and every interval [-r, r] and interval (-r, r) for r positive real.
- \(\mathcal{T}\) consists of \(\mathbb{R}, \emptyset\), and every interval [-n, n] and interval (-r, r) for n positive integer and r positive real.
Not a topology on \(\mathbb{R}\)
- \(\mathcal{T}\) consists of \(\mathbb{R}, \emptyset\), and every interval [-r, r] for r positive real. ▶ Why?
You can always find some infinite monotonic increasing sequence of real number that converges to a real value r, but never precisely reach r. The infinite union of sets derived from the sequence converges to \((-r,r)\notin\mathcal{T}\).
- \(\mathcal{T}\) consists of \(\mathbb{R}, \emptyset\), and every interval (-q, q) for q positive rational. ▶ Why?
You can find some infinite monotonic increasing sequence of rational number that converges to an irrational number, e.g. partial sums of the Maclaurin series representation of \(e\). The infinite union of sets derived from the sequence converges to \((-r,r)\notin\mathcal{T}\).