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Topology

Point-set topology foundations, compactness, connectedness, separation axioms, and an introduction to algebraic topology through the fundamental group and covering spaces at the CSIR NET level.

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01 Topological Spaces

Open sets, bases, subbases, product topology, subspace topology, and quotient topology.

◆ Definitions & Bases
Definition A topology on a set $X$ is a collection $\tau$ of subsets (called open sets) satisfying: (1) $\emptyset, X \in \tau$, (2) arbitrary unions of members of $\tau$ are in $\tau$, (3) finite intersections of members of $\tau$ are in $\tau$.
Basis for a Topology A collection $\mathcal{B}$ of open sets is a basis for $\tau$ if every open set is a union of members of $\mathcal{B}$. Equivalently, for every open $U$ and $x \in U$, there exists $B \in \mathcal{B}$ with $x \in B \subseteq U$.
Product Topology On $X = \prod_{\alpha} X_\alpha$, the product topology has basis $\{\prod_\alpha U_\alpha : U_\alpha \text{ open}, U_\alpha = X_\alpha \text{ for all but finitely many } \alpha\}$. This is the coarsest topology making all projections $\pi_\alpha: X \to X_\alpha$ continuous.
📝 Example
Describe the standard topology on $\mathbb{R}^2$ as a product topology.
$\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}$ with the product of the standard topologies on each factor. A basis consists of open rectangles $(a,b) \times (c,d)$. This generates the same topology as the metric topology from $d((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$, since open balls and open rectangles generate the same topology.
◆ Quotient Topology
Definition Given a surjection $p: X \to Y$, the quotient topology on $Y$ is $\{U \subseteq Y : p^{-1}(U) \text{ is open in } X\}$. This is the finest topology making $p$ continuous.

Key examples: $[0,1]/\{0 \sim 1\} \cong S^1$ (circle). $D^2 / \partial D^2 \cong S^2$ (sphere from collapsing the boundary of a disk). The torus $T^2 = [0,1]^2 / \sim$ with opposite sides identified.

02 Continuity & Homeomorphisms

Continuous maps between topological spaces, homeomorphisms, topological invariants, and embedding theorems.

◆ Continuous Maps
Definition $f: X \to Y$ is continuous if $f^{-1}(U)$ is open in $X$ for every open $U \subseteq Y$. Equivalently (for metric spaces), the $\varepsilon$-$\delta$ definition. A homeomorphism is a continuous bijection with continuous inverse.
Topological Invariants Properties preserved by homeomorphisms are called topological invariants. These include: compactness, connectedness, path-connectedness, Hausdorff property, first/second countability, and the fundamental group.
📝 Example
Show that $(0,1)$ and $\mathbb{R}$ are homeomorphic.
The map $f: (0,1) \to \mathbb{R}$ defined by $f(x) = \tan(\pi x - \pi/2)$ is a continuous bijection with continuous inverse $f^{-1}(y) = \frac{1}{\pi}\arctan(y) + \frac{1}{2}$. Hence $(0,1) \cong \mathbb{R}$.
📝 Example
Prove that $[0,1]$ and $(0,1)$ are not homeomorphic.
$[0,1]$ is compact but $(0,1)$ is not (the cover $\{(1/n, 1-1/n)\}$ has no finite subcover). Since compactness is a topological invariant, they cannot be homeomorphic. Alternatively: removing any point from $(0,1)$ disconnects it if and only if the point is not an endpoint, while $[0,1] \setminus \{0\} = (0,1]$ is connected.
03 Compactness

Open cover definition, Heine-Borel, Tychonoff's theorem, sequential and limit point compactness, and local compactness.

◆ Compactness & Key Theorems
Definition A topological space $X$ is compact if every open cover of $X$ has a finite subcover.
Heine-Borel Theorem A subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded.
Tychonoff's Theorem An arbitrary product $\prod_{\alpha} X_\alpha$ (with the product topology) is compact if and only if each $X_\alpha$ is compact. This is equivalent to the Axiom of Choice.
Properties of Compact Spaces
  • Closed subsets of compact spaces are compact
  • Compact subsets of Hausdorff spaces are closed
  • Continuous images of compact spaces are compact
  • A continuous bijection from a compact space to a Hausdorff space is a homeomorphism
Sequential vs Limit Point Compactness Sequentially compact: every sequence has a convergent subsequence. Limit point compact: every infinite subset has a limit point. In metric spaces, all three notions (compact, sequentially compact, limit point compact) coincide.
📝 Example
Is $\{0, 1\}^{\mathbb{N}}$ (infinite product of $\{0,1\}$ with discrete topology) compact?
Yes, by Tychonoff's theorem: each factor $\{0,1\}$ is compact (finite), so the product is compact. This space is homeomorphic to the Cantor set.
04 Connectedness

Connected and path-connected spaces, components, local connectedness, and the intermediate value theorem as a topological result.

◆ Connected Spaces
Definition A space $X$ is connected if it cannot be written as $X = U \cup V$ where $U, V$ are nonempty, disjoint, open sets. Equivalently, the only subsets that are both open and closed are $\emptyset$ and $X$.
Intermediate Value Theorem (Topological) If $f: X \to \mathbb{R}$ is continuous and $X$ is connected, then $f(X)$ is an interval (connected subset of $\mathbb{R}$).
Path-Connectedness $X$ is path-connected if for any $x, y \in X$, there exists a continuous map $\gamma: [0,1] \to X$ with $\gamma(0) = x$, $\gamma(1) = y$. Path-connected $\implies$ connected, but not conversely in general.
Connected $\not\Rightarrow$ Path-Connected The topologist's sine curve $S = \{(x, \sin(1/x)) : x > 0\} \cup \{(0,y) : -1 \leq y \leq 1\}$ is connected but not path-connected. It is the closure of the graph of $\sin(1/x)$.
📝 Example
Show that $\mathbb{R}^n \setminus \{0\}$ is connected for $n \geq 2$.
For $n \geq 2$: given $x, y \in \mathbb{R}^n \setminus \{0\}$, if the straight line segment from $x$ to $y$ avoids $0$, we have a path. If it passes through $0$, pick any $z$ not on the line through $x$ and $y$; then the segments $x \to z$ and $z \to y$ both avoid $0$ (since $0$ lies only on the line through $x$ and $y$, and $z$ is off this line). So $\mathbb{R}^n \setminus \{0\}$ is path-connected, hence connected.
◆ Components & Local Connectedness
Definition The connected components of $X$ are maximal connected subsets; they partition $X$ and are always closed. The path components are maximal path-connected subsets.

$X$ is locally connected at $x$ if every neighborhood of $x$ contains a connected open neighborhood. A space can be connected without being locally connected (topologist's sine curve).

05 Separation Axioms

The hierarchy T0 through T4, regularity, normality, Urysohn's lemma, and Tietze extension theorem.

◆ The Separation Hierarchy
Separation Axioms
  • $T_0$ (Kolmogorov): For distinct $x, y$, at least one has an open neighborhood not containing the other
  • $T_1$ (Frechet): For distinct $x, y$, each has an open neighborhood not containing the other. Equivalent to: singletons are closed
  • $T_2$ (Hausdorff): Distinct points have disjoint open neighborhoods
  • $T_3$ (Regular): $T_1$ + points and closed sets can be separated by open sets
  • $T_4$ (Normal): $T_1$ + disjoint closed sets can be separated by open sets
Urysohn's Lemma $X$ is normal if and only if for every pair of disjoint closed sets $A, B \subseteq X$, there exists a continuous function $f: X \to [0,1]$ with $f(A) = \{0\}$ and $f(B) = \{1\}$.
Tietze Extension Theorem If $X$ is normal and $A \subseteq X$ is closed, then every continuous $f: A \to \mathbb{R}$ extends to a continuous $\tilde{f}: X \to \mathbb{R}$. If $f$ is bounded, the extension can be chosen with the same bounds.
Key Facts
  • Metric spaces are normal (hence satisfy all $T_i$)
  • Compact Hausdorff spaces are normal
  • Every second countable regular space is metrizable (Urysohn metrization theorem)
06 Fundamental Group & Covering Spaces

Homotopy, the fundamental group, Van Kampen's theorem, covering spaces, and lifting properties.

◆ Fundamental Group
Definition Two paths $\gamma_0, \gamma_1: [0,1] \to X$ with the same endpoints are homotopic (rel endpoints) if there is a continuous $H: [0,1]^2 \to X$ with $H(t,0) = \gamma_0(t)$, $H(t,1) = \gamma_1(t)$, $H(0,s) = x_0$, $H(1,s) = x_1$. The fundamental group $\pi_1(X, x_0)$ is the set of homotopy classes of loops based at $x_0$ under concatenation.
Key Computations
  • $\pi_1(\mathbb{R}^n) = 0$ (contractible spaces have trivial fundamental group)
  • $\pi_1(S^1) \cong \mathbb{Z}$ (generated by the loop $t \mapsto e^{2\pi i t}$)
  • $\pi_1(T^2) \cong \mathbb{Z} \times \mathbb{Z}$ (torus)
  • $\pi_1(S^n) = 0$ for $n \geq 2$ (higher spheres are simply connected)
Van Kampen's Theorem If $X = U_1 \cup U_2$ with $U_1, U_2, U_1 \cap U_2$ path-connected and open, then $\pi_1(X) \cong \pi_1(U_1) *_{\pi_1(U_1 \cap U_2)} \pi_1(U_2)$ (amalgamated free product). In particular, if $U_1 \cap U_2$ is simply connected, $\pi_1(X) \cong \pi_1(U_1) * \pi_1(U_2)$ (free product).
📝 Example
Compute $\pi_1(S^1 \vee S^1)$ (wedge of two circles).
Take $U_1$ = neighborhood of first circle, $U_2$ = neighborhood of second circle, chosen so that $U_1 \cap U_2$ is a contractible neighborhood of the wedge point. By Van Kampen: $\pi_1(S^1 \vee S^1) \cong \pi_1(S^1) * \pi_1(S^1) \cong \mathbb{Z} * \mathbb{Z}$, the free group on two generators.
◆ Covering Spaces
Definition A covering space of $X$ is a space $\tilde{X}$ with a map $p: \tilde{X} \to X$ such that every $x \in X$ has an open neighborhood $U$ with $p^{-1}(U) = \bigsqcup_\alpha V_\alpha$ where each $V_\alpha$ is mapped homeomorphically onto $U$ by $p$.
Lifting & Classification Key properties: (1) Paths lift uniquely given a starting point. (2) Homotopies of paths lift. (3) The number of sheets equals $[\pi_1(X) : p_*\pi_1(\tilde{X})]$. (4) Connected covering spaces of $X$ (up to equivalence) correspond to conjugacy classes of subgroups of $\pi_1(X)$.

Examples: $p: \mathbb{R} \to S^1$ via $p(t) = e^{2\pi i t}$ is the universal cover of $S^1$ (infinite-sheeted). $p: S^1 \to S^1$ via $z \mapsto z^n$ is an $n$-sheeted cover.

★ Key Takeaways
✍ Practice Problems
Problem 1
Show that $\mathbb{R}$ with the cofinite topology (closed sets are finite sets and $\mathbb{R}$) is $T_1$ but not Hausdorff.
Show Solution ▼
$T_1$: For distinct $x, y$, the set $\mathbb{R} \setminus \{y\}$ is open (its complement $\{y\}$ is finite), contains $x$, and excludes $y$. Not Hausdorff: any two nonempty open sets $U, V$ have finite complements, so $U \cap V = \mathbb{R} \setminus (U^c \cup V^c)$ has complement that is a finite union of finite sets, hence finite. Since $\mathbb{R}$ is infinite, $U \cap V \neq \emptyset$. Thus no two distinct points can be separated by disjoint open sets.
Problem 2
Prove that a continuous bijection $f: X \to Y$ from a compact space to a Hausdorff space is a homeomorphism.
Show Solution ▼
We need to show $f^{-1}$ is continuous, i.e., $f$ is a closed map. Let $C \subseteq X$ be closed. Since $X$ is compact, $C$ is compact. Since $f$ is continuous, $f(C)$ is compact. Since $Y$ is Hausdorff, compact subsets are closed, so $f(C)$ is closed in $Y$. Hence $f$ maps closed sets to closed sets, so $f^{-1}$ is continuous.
Problem 3
Show that $\mathbb{Q}$ (with the subspace topology from $\mathbb{R}$) is totally disconnected: its only connected subsets are singletons.
Show Solution ▼
Let $A \subseteq \mathbb{Q}$ contain two distinct points $p < q$. Choose an irrational $r$ with $p < r < q$. Then $A = (A \cap (-\infty, r)) \cup (A \cap (r, \infty))$ is a partition of $A$ into two nonempty relatively open sets (nonempty since $p$ is in the first, $q$ in the second). So $A$ is disconnected. Hence every connected subset has at most one point.
Problem 4
Use the fundamental group to prove that $\mathbb{R}^2$ is not homeomorphic to $\mathbb{R}^n$ for $n \neq 2$.
Show Solution ▼
If $f: \mathbb{R}^2 \to \mathbb{R}^n$ is a homeomorphism, then $\mathbb{R}^2 \setminus \{p\} \cong \mathbb{R}^n \setminus \{f(p)\}$. For $n = 1$: $\mathbb{R} \setminus \{0\}$ is disconnected but $\mathbb{R}^2 \setminus \{0\}$ is connected; contradiction. For $n \geq 3$: $\pi_1(\mathbb{R}^2 \setminus \{0\}) \cong \pi_1(S^1) \cong \mathbb{Z}$ but $\pi_1(\mathbb{R}^n \setminus \{0\}) \cong \pi_1(S^{n-1}) = 0$ for $n \geq 3$ (since $S^{n-1}$ is simply connected for $n-1 \geq 2$). Since $\mathbb{Z} \not\cong 0$, contradiction.
Problem 5
Find all connected $2$-sheeted covering spaces of $S^1 \vee S^1$ (up to equivalence).
Show Solution ▼
$\pi_1(S^1 \vee S^1) \cong F_2 = \langle a, b \rangle$ (free group on 2 generators). A connected $2$-sheeted cover corresponds to a subgroup of index $2$ in $F_2$, which is the same as a surjective homomorphism $F_2 \to \mathbb{Z}/2\mathbb{Z}$ (the kernel being the subgroup). Such homomorphisms are determined by $\phi(a), \phi(b) \in \{0, 1\}$, but $\phi$ must be surjective, so at least one of $\phi(a), \phi(b)$ is $1$. This gives $3$ possibilities: $(1,0), (0,1), (1,1)$. Hence there are exactly 3 connected $2$-sheeted covers.
🎯 Test Your Understanding
1. Which of the following subsets of $\mathbb{R}$ is compact?
A $(0, 1)$
B $[0, 1]$
C $\mathbb{Z}$
D $\mathbb{R}$
2. The topologist's sine curve is:
A Both connected and path-connected
B Connected but not path-connected
C Path-connected but not connected
D Neither connected nor path-connected
3. $\pi_1(S^2)$ is:
A Trivial (the zero group)
B $\mathbb{Z}$
C $\mathbb{Z}_2$
D $\mathbb{Z} \times \mathbb{Z}$
4. A compact Hausdorff space is always:
A Metrizable
B Normal
C Second countable
D Path-connected
5. The number of path components of $\mathbb{R} \setminus \mathbb{Q}$ (the irrationals) is:
A 1
B 2
C Countably infinite
D Uncountably infinite