Introduction to Topology

Topology is the branch of mathematics that studies properties of spaces that are preserved under continuous deformations. Two objects are considered topologically equivalent if one can be continuously deformed into the other without tearing or gluing.

The classic informal example: a coffee mug is topologically equivalent to a donut (torus). Both have exactly one hole, and one can be smoothly deformed into the other.

Formally, topology generalises the notions of closeness, continuity, and convergence from analysis and geometry, stripping away the rigid structure of distance and angles.

Topology arises by definition from several directions:

• Analysis: The $\varepsilon$-$\delta$ definition of continuity in $\mathbb{R}^n$ depends only on open sets, not on the specific metric. Topology abstracts this to general spaces.
• Geometry: Many geometric properties (e.g., the number of holes in a surface) are invariant under continuous deformations.
• Algebra: Topological spaces carry algebraic invariants (fundamental group, homology groups) that classify spaces up to topological equivalence.

Key figures include Euler (Königsberg bridges, 1736), Riemann (Riemann surfaces, 1850s), Cantor (set theory, 1870s—80s), Poincaré (algebraic topology, 1890s), Hausdorff (metric and topological spaces, 1914), and Brouwer (fixed point theorem, 1911).

A topological space is a set $X$ together with a collection $\mathcal{T}$ of subsets of $X$ (called open sets) satisfying:

1. $\varnothing$ and $X$ are in $\mathcal{T}$.
2. Arbitrary unions of elements of $\mathcal{T}$ are in $\mathcal{T}$.
3. Finite intersections of elements of $\mathcal{T}$ are in $\mathcal{T}$.

The collection $\mathcal{T}$ is called a topology on $X$. A set is closed if its complement is open.

Example 1.1 (Standard Topology on $\mathbb{R}$). The standard topology on $\mathbb{R}$ has open sets as arbitrary unions of open intervals $(a, b)$. This gives the usual notion of continuity.

Example 1.2 (Discrete Topology). On any set $X$, let every subset be open. This is the finest topology: it makes every function $f : X \to Y$ continuous.

Example 1.3 (Indiscrete Topology). On any set $X$, let only $\varnothing$ and $X$ be open. This is the coarsest topology. Only constant functions are continuous from this space.

Example 1.4 (Cofinite Topology). On an infinite set $X$, let $U \subseteq X$ be open if $U = \varnothing$ or $X \setminus U$ is finite. This topology satisfies only the finite intersection property for open sets.

A basis $\mathcal{B}$ for a topology on $X$ is a collection of subsets of $X$ such that:

1. For each $x \in X$, there exists $B \in \mathcal{B}$ with $x \in B$.
2. If $x \in B_1 \cap B_2$ for $B_1, B_2 \in \mathcal{B}$, then there exists $B_3 \in \mathcal{B}$ with $x \in B_3 \subseteq B_1 \cap B_2$.

The topology generated by $\mathcal{B}$ consists of all unions of elements of $\mathcal{B}$.

Example 1.5. The collection of all open intervals $(a, b)$ in $\mathbb{R}$ is a basis for the standard topology. The collection of all open balls $B_r(x)$ in a metric space forms a basis.

A function $f : X \to Y$ between topological spaces is continuous if for every open set $V \subseteq Y$, the preimage $f^{-1}(V) \subseteq X$ is open.

Proposition 1.1 (Equivalent Characterizations). The following are equivalent:

1. $f$ is continuous.
2. For every closed set $C \subseteq Y$, $f^{-1}(C) \subseteq X$ is closed.
3. For every $x \in X$ and every neighborhood $V$ of $f(x)$, there exists a neighborhood $U$ of $x$ such that $f(U) \subseteq V$.

Proposition 1.2. The composition of continuous functions is continuous. The identity map $\mathrm{id}_X : X \to X$ is continuous.

Example 1.6. Any function from a discrete space to any topological space is continuous. A function from any topological space to an indiscrete space is continuous.

Definition. A homeomorphism is a bijection $f : X \to Y$ such that both $f$ and $f^{-1}$ are continuous. Homeomorphic spaces are considered topologically equivalent.

For a subset $A \subseteq X$ of a topological space:

• The closure $\overline{A}$ is the intersection of all closed sets containing $A$.
• The interior $\mathrm{int}(A)$ is the union of all open sets contained in $A$.
• The boundary $\partial A = \overline{A} \setminus \mathrm{int}(A)$.

Proposition 1.3 (Properties).

• $A \subseteq \overline{A}$, and $\overline{A} = A$ iff $A$ is closed.
• $\mathrm{int}(A) \subseteq A$, and $\mathrm{int}(A) = A$ iff $A$ is open.
• $\overline{\overline{A}} = \overline{A}$, $\mathrm{int}(\mathrm{int}(A)) = \mathrm{int}(A)$.

Example 1.7. In $\mathbb{R}$ with the standard topology:

• $\overline{(0, 1)} = [0, 1]$, $\mathrm{int}([0, 1]) = (0, 1)$.
• $\overline{\mathbb{Q}} = \mathbb{R}$ (the rationals are dense), $\mathrm{int}(\mathbb{Q}) = \varnothing$.
• $\partial((0, 1]) = \{0, 1\}$.

Subspace topology. If $Y \subseteq X$, the subspace topology on $Y$ is $\mathcal{T}_Y = \{U \cap Y : U \in \mathcal{T}_X\}$. A set is open in $Y$ if it is the intersection of $Y$ with an open set in $X$.

Product topology. For two topological spaces $X$ and $Y$, the product topology on $X \times Y$ has basis $\{U \times V : U \text{ open in } X,\ V \text{ open in } Y\}$. Projection maps $\pi_X : X \times Y \to X$ and $\pi_Y : X \times Y \to Y$ are continuous.

Example 1.8. The product topology on $\mathbb{R}^2 = \mathbb{R} \times \mathbb{R}$ coincides with the standard Euclidean topology. A basis is given by open rectangles $(a, b) \times (c, d)$.

A metric space $(X, d)$ induces a topology where $U$ is open if for every $x \in U$ there exists $\varepsilon > 0$ such that $B_\varepsilon(x) \subseteq U$. The open balls form a basis.

Proposition 1.4. Every metric space is Hausdorff: for any distinct $x, y \in X$, there exist disjoint open sets $U, V$ with $x \in U$, $y \in V$.

Example 1.9. The discrete metric ($d(x, y) = 1$ for $x \neq y$) induces the discrete topology. The usual metric on $\mathbb{R}^n$ induces the standard topology.

Problem 1. Show that the collection $\mathcal{B} = \{[a, b) : a < b\}$ forms a basis for a topology on $\mathbb{R}$ (the lower-limit topology, or Sorgenfrey line).

Solution. Every $x \in \mathbb{R}$ belongs to $[x, x+1) \in \mathcal{B}$. If $x \in [a, b) \cap [c, d)$, then $x$ is in $[\max(a, c), \min(b, d))$ which is in $\mathcal{B}$. $\blacksquare$

Problem 2. Prove that $f : X \to Y$ is continuous iff for every $A \subseteq X$, $f(\overline{A}) \subseteq \overline{f(A)}$.

Solution. Assume $f$ is continuous. Take $y \in f(\overline{A})$, so $y = f(x)$ with $x \in \overline{A}$. For any open $V \ni y$, $f^{-1}(V)$ is an open neighborhood of $x$, so $f^{-1}(V) \cap A \neq \varnothing$. Thus $V \cap f(A) \neq \varnothing$, so $y \in \overline{f(A)}$. Conversely, if the condition holds, let $C \subseteq Y$ be closed. Then $f(\overline{f^{-1}(C)}) \subseteq \overline{f(f^{-1}(C))} \subseteq \overline{C} = C$, so $\overline{f^{-1}(C)} \subseteq f^{-1}(C)$, hence $f^{-1}(C)$ is closed. $\blacksquare$

1. Show that the standard topology on $\mathbb{R}$ is the same as the metric topology from $d(x, y) = |x-y|$.
2. Prove that continuous functions preserve compactness: if $f$ is continuous and $K$ is compact, then $f(K)$ is compact.
3. Show that every closed subset of a compact space is compact.
4. Find the closure, interior, and boundary of $\mathbb{Q} \subseteq \mathbb{R}$.
5. Prove that $X$ is Hausdorff iff the diagonal $\Delta = \{(x, x) : x \in X\} \subseteq X \times X$ is closed.