Topology summary

Continuous map

A function f:XYf:X \to Y is said to be continuous if for each open subset VV of YY, f1(V)f^{-1}(V) is an open set of XX.

Let XX, and YY be topologies. The followings are equivalent

  1. f:XYf:X \to Y is continuous.
  2. for every closed set BB of YY, f1(B)f^{-1}(B) is closed in XX.
  3. f(A)f(A)f(\overline A) \subseteq \overline {f(A)} and f1(B)f1(B)\overline {f^{-1}(B)} \subseteq f^{-1}(\overline B) for AXA \subset X, BYB \subset Y.
  4. For each xXx \in X and each neighborhood VV of f(x)f(x), there is a neighborhood UU of xx such that f(U)Vf(U) \subset V
  5. for a basis BB of YY, f1(B)f^{-1}(B) is open in XX.


Let XX and YY be topological spaces; let f:XYf:X \to Y be a one-to-one, onto, continuous, and the inverse function is continuous, then ff is a Homeomorphism.
Let SnS^n be a n-dimensional sphere, p=(0,,0,1)p = (0,\,\ldots,\,0, 1) be the north pole. h:Sn{p}Rnh:S^n - \{p\} \to \mathbb R^nbe the Stereographic projection is defined.

Space-filling curve

Let I=[0,1]I=[0,1], there exists f:II×If: I \to I\times I such that this map fill up the entire square I2I^2.

For limmfmf\displaystyle\lim_{m \to \infty} f_m \to f and d:(I2,I2)Rd:(I^2, I^2) \to R is distance, limnd(fn,f)0\displaystyle\lim_{n \to \infty}d(f_n,f) \to 0. and then since fnf_n is a continuous, ff is continuous.

A metric function

  1. d:X×XRd: X \times X \to \mathbb R
  2. d(x,y)0d(x,y) \ge 0, equality holds iff x=yx=y
  3. d(x,y)=d(y,x)d(x,y) = d(y,x)
  4. d(x,y)+d(z,y)d(x,y)d(x,y) + d(z,y) \ge d(x,y)

a set with a metric is called a metric space.

a distance between two functions is d(f,g)=supt[0,1]f(t)g(t)d(f,g) = \displaystyle\sup_{t \in [0,1]}|f(t) - g(t)|

if AA is a subset of XX, the real-valued function d(x,A)d(x,A) is defined to be the infimum of d(x,A)d(x,A) where aAa \in A and is continuous.

Tietze extension theorem.

Any real valued continuous function on a closed subset of a metric space may be extended to a real-valued continuous function on the whole space.

Compact spaces

Let A\mathcal A be a collection of open subsets of space XX. A\mathcal A is called an open cover of XX such that A=X\bigcup \mathcal A = X. and A\mathcal A' is called a subcover of A\mathcal A if A\mathcal A' is a subcollection of A\mathcal A and A=X\bigcup \mathcal A' =X.
A space XX is said to be compact, if all open cover of XX has a finite subcover.

Other defintion of a compactness in terms of closed subsets

Heine-Borel theorem

A closed interval of the real line is compact.


A space XX is connectedness if when we write XX as union ABA \cup B of two nonempysubsets, then A¯B or AB¯\bar A \cap B \neq \emptyset \text{ or } A \cap \bar B \neq \emptyset. in other words, XX is not seperated by disjoint nonempty open subsets.


  1. A indiscrete topology is a connected space.
  2. The real line is a connected space.
  3. A rational C\mathbb C is not a connected space.


a component of topology XX is a maximal connected subset of XX. so if a space is not connected then it is seperated by any components.

joinning poins by path

Given x,yAx, y \in A, A path in the topological space AA is a continuous function f:[a,b]Af:[a, b] \to A such that f(a)=xf(a)=x, f(b)=yf(b)=y. and A space AA is said to path connected if any two of its points can be joined by a path.


  1. the continuous image of path-connected space is path-connected.
  2. SnS^n(n sphere) is path-connected for n>0n > 0.
  3. the product of two path-connected space is path-connected.
  4. any indiscrete space is path-connected.

Homotopic map

Let X,YX, Y be topological spaces.Let f,g:XYf,g:X \to Y are continuous. ff is homotopic to gg if there exists F:X×[a,b]YF:X \times [a, b] \to Y such that F(x,0)=f(x)F(x,0) = f(x) and F(x,1)=g(x)F(x,1)=g(x) for all points xXx \in X.
and this continuous map FF be called homotopy from ff to gg .