Summary

Concept	Key Idea
Topological space	Set + open sets satisfying the three axioms
Closed sets	Complements of open sets; finite unions, arbitrary intersections
Closure / interior / boundary	$\overline{A}$ , $\operatorname{int}(A)$ , $\partial A = \overline{A} \setminus \operatorname{int}(A)$
Continuity	$f^{-1}(\text{open})$ is open
Homeomorphism	Bijective continuous map with continuous inverse
Compactness	Every open cover has a finite subcover
Heine–Borel	In $\mathbb{R}^n$ : compact $\Leftrightarrow$ closed and bounded
Connectedness	No separation into two disjoint nonempty open sets
Path-connectedness	Any two points joined by a continuous path
Metric space	Set + distance function satisfying the three axioms
Completeness	Every Cauchy sequence converges
Banach fixed point	Contractions on complete metric spaces have unique fixed points
$T_0$ – $T_4$	Increasingly strong separation axioms
Fundamental group $\pi_1$	Homotopy classes of loops; a topological invariant
Euler characteristic $\chi$	$V - E + F$ ; classifies compact surfaces

Worked Examples

Example 1: Determining if a Collection is a Topology

Problem: Is the collection $\tau = \{\emptyset, \{a\}, \{b\}, \{a,b,c\}\}$ a topology on $X = \{a,b,c\}$ ? Solution: Check axioms: (1) $\emptyset$ and $X$ are in $\tau$ . (2) Finite unions: $\{a\} \cup \{b\} = \{a,b\}$ , which is NOT in $\tau$ . Therefore $\tau$ is not a topology. To fix it, we would need to include $\{a,b\}$ .

Example 2: Continuous Function Proof

Problem: Show that the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$ is continuous with respect to the standard topology. Solution: Let $U$ be an open set in $\mathbb{R}$ . $f^{-1}(U) = \{x : x^2 \in U\}$ . For any open interval $(a, b)$ with $a \geq 0$ , the preimage is $(-\sqrt{b}, -\sqrt{a}) \cup (\sqrt{a}, \sqrt{b})$ , which is a union of open intervals (open). For negative intervals, the preimage is empty or $\mathbb{R}$ . Since the preimage of any basis element is open, $f$ is continuous.

Cross-References

Topic	Link
Abstract Algebra	View

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