Notes

Description: Topology definitions

Content:

Standard definitions from point set topology. In this note we give standard definitions from point set topology which do not occur elsewhere.

Definition:

  1. A pseudometric on \(X\) is a function \(d: X\times X \to X\) such that
    • \(\forall x,y\in X\), \(d(x,y)\ge 0\)
    • \(d(x,y) + d(y,z) \ge d(x,z)\)
    • \(d(x,x) = 0\).
    In the following \((X,d)\) is a pseudometric space.
  2. \(S(y,r)\) is the ball of radius \(r\) centered at \(y\).
  3. \((X,d)\) is complete if every Cauchy net converges. (In Note 28, a metric space is defined to be complete or Frechet complete if Cauchy sequences converge. When this definition of complete is intended we will refer to Note 28.)
  4. \((X,d)\) is totally bounded (or TBd) if for every positive real number \(r\) there is a finite set \(Y\) such that \(X = \bigcup_{y\in Y} S(y,r)\).
  5. \((X,d)\) is sequentially bounded if every sequence in \(X\) has a Cauchy subsequence.
  6. \((x_\lambda)_{\lambda\in\Lambda}\) is a Cauchy net in \((X,d)\) if \((\forall r\in\Bbb R)(\exists\lambda_0\in\Lambda) (\forall\lambda_1,\lambda_2\in\Lambda)\)(If \(\lambda_1,\lambda_2\ge \lambda_0\), then \(x_{\lambda_1},x_{\lambda_2}\in S(x,r)\)for some \(x\in X\)).
  7. \(\cal F\) is a Cauchy filter in \((X,d)\) if \()(\forall r\in\Bbb R)(\exists x\in X)(S(x,r)\in\cal F)\).

The diagram below from Bentley/Herrlich [1998] gives some relationships in ZF\(^0\) between various forms of compactness for pseudometric spaces.  (In the diagram ``H-B'' means `` Heine-Borel compact'', ``A-U'' means ``Alexandroff-Urysohn compact'', ``C-TBd'' means  ``complete and totally bounded, ``SEQ'' means ``sequentially compact'', ``W'' means ``Weierstrass compact'' (see Note 6), ``\(A\Rightarrow B\)'' means ``\(A\) implies \(B\), but \(B\) does not imply \(A\)'', ``\(A \not\Leftrightarrow B\)'' means ``\(A\) does not imply \(B\) and \(B\) does not imply \(A\)'', and ``\(A\Leftrightarrow B\)'' means ``\(A\) implies \(B\) and \(B\) implies \(A\)''.)

\[ \begin{matrix} \boxed{\text{A-U}} & \kern5pt\not\kern-5pt\Longleftrightarrow & \boxed{\text{H-B}}\\ \Downarrow & & \Updownarrow \\ \boxed{\text{W}} & \Longleftarrow & \boxed{\text{H-B}} \\ \Downarrow & & \Downarrow \\ \boxed{\text{SEQ}} & \Longleftarrow & \boxed{\text{C-TBd}} \end{matrix} \]

Definition: Assume \((X,T)\) is a topological space.

  1. A zero-set in \(X\) is a set of the form \(\{x\in X : f(x) = 0 \}\) where \(f:X\to [0,1]\) is continuous.
  2. A closed (respectively open, clopen, z-) filter is a filter in the lattice of all closed (respectively open, clopen, zero-) sets in \(X\).
  3. A filter \(\cal F\) in a lattice \(L\) is prime if for all \(A\) and \(B\) in \(L\), if \(A\cup B\in\cal F\) then either \(A\in \cal F\) or \(B\in\cal F\).
  4. A subset \(\cal B\) of a filter in \(\cal P(X)\) is a filter base for a filter \(\cal F\) if \(\forall B_1, B_2 \in \cal B\), there is a \(B_3\in \cal B\) such that \(B_3\subseteq B_1\cap B_2\) and \(\cal F = \{C : B\subseteq C\) for some \(B \in \cal B\}\).
  5. A free filter is a non-principal filter.

Definition: Let \(F\) be a family of functions from \(X\) to \(Y\) where \(X\) and \(Y\) are topological spaces.  The compact-open topology on \(F\) is the topology with subbase consisting of all sets of the form \(\{f\in F : f[K] \subseteq U\}\) where \(K\) is a compact subset of \(X\) and \(U\) is an open subset of \(Y\).

Definition: A set \(F\) of functions from \(X\) to \(Y\), where \(Y\) is a metric space is equicontinuous if \(\forall x\in X\), \(\forall \epsilon >0\), \(\exists\) a neighborhood \(U\) of \(x\) such that \(\forall f\in F\) and \(\forall y\in U\), \(d(f(x),f(y)) < \epsilon\).

Definition: If \((f_n)_{n\in\omega}\) is a sequence of continuous functions from a topological space \(X\) to a topological space \(Y\) and \(f: X\to Y\), then \((f_n)_{n\in\omega}\) converges continuously to \(f\) provided \(\forall x\in X\), and for all sequences \((x_n)_{n\in\omega}\) of elements of \(X\) such that \((x_n)_{n\in\omega}\to x\), \((f(x_n))_{n\in\omega} \to f(x)\).

Howard-Rubin number: 10

Type: Definitions

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