拓撲筆記

Chap 1 Point Set Topology

Lec1

Def: \(T\) topology of X if: (1) \(\empty, X\in T\) (2) ( union of elts in \(T\)) \(\in T\) (3) \(a, b\in T\), then \(a\cap b \in T\).

Notation: elts of \(T\) open; \((X, T)\) topological space; \(C \subset X\) closed if \(X/C\) open; \(A\) neighborhood of \(x\) if \(x \in B \subset A, B\in T\).

Fact: \(A\) open iff \(A\) is nbhd of all its elts.

Lec2

Def: \(f: X\to Y\) continuous if for any \(U\subset Y\) open, \(f^{-1}(U)\) open in \(X\).

Prop: TFAE

(1) \(f\) continuous

(2) \(f^{-1}(C)\) closed in \(X\) for any \(C\) closed in \(Y\)

(3) any \(x\in X\), any nbhd \(V\) of \(f(x)\), \(f^{-1}(V)\) is a nbhd of \(x\)

(Proof: (1) equiv to (2) is trivial. (1) to (3) is trivial. (3) to (1) since the inverse image is nbhd of every elt. )

Example: continuous on standard topology is the usual meaning of continuous.

Def: f homeomorphism if \(f, f^{-1}\) continuous and bijective.

Def: Induced topology \(T_A\) on \(A\sub X\) are the elements \(v\cap A\) with \(v\in T\).

ALS 1

Def. \((X, T_1), (X, T_2)\) topological spaces with \(T_1\subset T_2\), then we call \(T_1\) coarser than \(T_2\), \(T_2\) finer than \(T_1\).

Def. interior \(A^\circ\) are the elements with \(A\) as nhbd

​ closure \(\overline A\) are the elements \(x\) such that \(X/A\) is not a nbhd of \(x\).