Topology Domain Guide

Concise answer

Topology is the study of which sets are declared open, and everything else follows. Continuity becomes 'preimages of open sets are open', which is why it needs no distances. Compactness and connectedness are the two properties a continuous map always carries forward; openness, closedness, and boundedness are not. Knowing which side of that line a property falls on answers the majority of exam items directly.

Definitions

Topology: A collection of subsets of $X$ containing $\emptyset$ and $X$ and closed under arbitrary unions and finite intersections. Its members are the open sets.
Closure, interior, boundary: $\overline{A}$ is the smallest closed set containing $A$ ; $A^{\circ}$ is the largest open set inside $A$ ; $\partial A = \overline{A} \setminus A^{\circ}$ .
Continuity (topological): $f$ is continuous when $f^{-1}(U)$ is open for every open $U$ . In a metric space this is equivalent to the $\varepsilon$ – $\delta$ definition.
Homeomorphism: A continuous bijection with continuous inverse. Continuity of the inverse is a separate requirement, not a consequence.
Compact: Every open cover has a finite subcover. In $\mathbb{R}^n$ this is equivalent to closed and bounded; in a general metric space it is equivalent to sequential compactness.
Connected: Not the union of two disjoint nonempty relatively open sets — equivalently, the only subsets that are both open and closed are $\emptyset$ and the space itself.
Hausdorff: Distinct points can be separated by disjoint open sets. Every metric space is Hausdorff, which is why limits there are unique.

Intuition

Once you can say which sets are open, you can say which functions are continuous, which sequences converge, and which sets are 'close' to which points — all without ever measuring a distance. That is the entire economy of topology: it keeps the notions that survive stretching and drops the ones that depend on size.

Compactness is best read as 'no escape to infinity and no escape through a missing boundary point'. Both failures are visible in $\mathbb{R}$ : $[0,\infty)$ escapes to infinity, and $(0,1]$ escapes toward the missing endpoint $0$ . Heine–Borel simply says those are the only two ways to fail in Euclidean space.

The single most useful mental table is what a continuous map does to a property. It carries compactness forward, and connectedness forward. It does not carry openness, closedness, or boundedness forward. Every corollary you know — the extreme value theorem, the intermediate value theorem — is one row of that table read aloud.

Concept walkthrough

The axioms have a deliberate asymmetry: arbitrary unions of open sets are open, but only finite intersections are. The reason is visible immediately — $\bigcap_{n \ge 1} (-1/n, 1/n) = \{0\}$ , an infinite intersection of open intervals that is not open. Dually, arbitrary intersections of closed sets are closed but only finite unions are. Exam items exploit this asymmetry constantly, usually by offering an infinite family and asking what the union or intersection must be.

'Open' and 'closed' are not opposites and not exclusive. In $\mathbb{R}$ , $[0,1)$ is neither; $\emptyset$ and $\mathbb{R}$ are both. A set that is both open and closed is called clopen, and a space is connected exactly when it has no clopen subsets besides the trivial two. Closure, interior, and boundary are the tools for handling such questions concretely: for $A = \mathbb{Q} \cap [0,1]$ the interior is empty, the closure is $[0,1]$ , and the boundary is all of $[0,1]$ — a good sanity check on any general claim you are tempted to make.

Continuity is where topology pays for itself. Defining $f$ continuous as 'the preimage of every open set is open' needs no distances and immediately gives the composition rule and the behaviour of restrictions. Note the direction: preimages, not images. A continuous map need not send open sets to open sets — $f(x) = x^2$ maps the open interval $(-1,1)$ onto $[0,1)$ , which is not open — and a map that does send open sets to open sets is called open, a separate property.

Compactness is the domain's centre of gravity. The definition is covering-based: every open cover has a finite subcover. In $\mathbb{R}^n$ , Heine–Borel reduces it to two checks, closed and bounded, and that reduction is what most items rely on. Outside $\mathbb{R}^n$ it genuinely fails: give any infinite set the discrete metric ( $d(x,y) = 1$ for $x \ne y$ ) and it is closed and bounded, yet the cover by singleton balls has no finite subcover. Two structural facts are worth holding: a closed subset of a compact space is compact, and a compact subset of a Hausdorff space is closed.

What compactness buys is stated as preservation. If $f$ is continuous and $K$ is compact then $f(K)$ is compact; specialising to $f : K \to \mathbb{R}$ , the image is closed and bounded and therefore contains its supremum and infimum — that is the extreme value theorem. Connectedness works the same way: if $C$ is connected then $f(C)$ is connected, and the connected subsets of $\mathbb{R}$ are exactly the intervals, so a continuous function on an interval takes every intermediate value. Deriving both theorems this way is cheaper than memorising them and makes it obvious which hypothesis each one needs.

The failures are as examinable as the theorems. Closedness is not preserved: $f(x) = 1/(1 + x^2)$ maps the closed set $\mathbb{R}$ onto $(0,1]$ . Boundedness is not preserved by continuous maps on unbounded domains. A continuous bijection need not be a homeomorphism: $t \mapsto (\cos t, \sin t)$ on $[0, 2\pi)$ is a continuous bijection onto the unit circle whose inverse is discontinuous at $(1,0)$ — though on a compact domain into a Hausdorff space, a continuous bijection is automatically a homeomorphism. Finally, connected does not imply path-connected; the topologist's sine curve is the standard witness.

After this page, you should be able to

Apply the topology axioms, including the asymmetry between arbitrary unions and finite intersections.
Use the preimage characterisation of continuity in place of $\varepsilon$ – $\delta$ when no metric is available.
Compute closure, interior, and boundary for standard subsets of $\mathbb{R}$ and $\mathbb{R}^2$ .
Decide compactness by Heine–Borel in $\mathbb{R}^n$ and by open covers or sequences elsewhere.
State which properties continuous maps preserve — compactness and connectedness — and produce a counterexample for the ones they do not.
Derive the extreme and intermediate value theorems as consequences of preservation, rather than memorising them separately.

Formulas and assumptions

Topology axioms

\emptyset, X \in \tau; \quad \bigcup_{\alpha} U_{\alpha} \in \tau; \quad U_1 \cap \cdots \cap U_n \in \tau

Variables

tau: the collection of open sets
X: the underlying set

Assumptions

The intersection clause is finite only: the intersection of all (-1/n, 1/n) is {0}, which is not open.
Closed sets satisfy the dual statements: arbitrary intersections, finite unions.

Continuity by preimages

f \text{ continuous} \iff f^{-1}(U) \text{ open for every open } U \iff f^{-1}(C) \text{ closed for every closed } C

Variables

f: a map between topological spaces

Assumptions

The condition is on preimages, not images; continuous maps need not be open or closed maps.
In metric spaces this is equivalent to the epsilon-delta definition.

Heine–Borel and its limits

K \subseteq \mathbb{R}^{n}: \ K \text{ compact} \iff K \text{ closed and bounded}

Variables

K: a subset of the space

Assumptions

The Euclidean equivalence is special to R^n; an infinite discrete metric space is closed and bounded but not compact.
Compactness and sequential compactness agree in metric spaces but not in general topological spaces.

What continuous maps preserve

f(K) \text{ compact}, \quad f(C) \text{ connected}; \qquad f(\text{closed}) \not\Rightarrow \text{closed}

Variables

K: a compact set
C: a connected set

Assumptions

f must be continuous and defined on the whole set in question.
The extreme and intermediate value theorems are the two preservation statements specialised to real-valued f.

Compactness and closedness interact

C \subseteq K \text{ closed},\ K \text{ compact} \Rightarrow C \text{ compact}; \qquad K \subseteq X \text{ compact},\ X \text{ Hausdorff} \Rightarrow K \text{ closed}

Variables

K: a compact set
X: the ambient space

Assumptions

The second statement needs the Hausdorff hypothesis; without it a compact set need not be closed.
Every metric space is Hausdorff, so both hold automatically in metric settings.

Connectedness in the real line

C \subseteq \mathbb{R} \text{ connected} \iff C \text{ is an interval}; \qquad \text{path-connected} \Rightarrow \text{connected}

Variables

C: a subset of the real line

Assumptions

The converse implication fails; the topologist's sine curve is connected but not path-connected.
Combining this with preservation of connectedness gives the intermediate value theorem.

Worked example

Compactness is preserved; closedness is not

The continuous image of a compact set is compact. Does the same statement hold with 'compact' replaced by 'closed'? Decide, and justify the decision.

1Take $f : \mathbb{R} \to \mathbb{R}$ with $f(x) = \dfrac{1}{1 + x^2}$ . It is a quotient of polynomials whose denominator never vanishes, so it is continuous on all of $\mathbb{R}$ .
2The domain $\mathbb{R}$ is closed in $\mathbb{R}$ : its complement is the empty set, which is open. So the hypothesis 'closed domain, continuous map' is satisfied.
3Compute the image. Since $1 + x^2 \ge 1$ with equality only at $x = 0$ , the largest value is $f(0) = 1$ . Every value is strictly positive, and $f(x) \to 0$ as $|x| \to \infty$ without ever reaching $0$ . So $f(\mathbb{R}) = (0, 1]$ .
4 $(0,1]$ is not closed: taking $x_n = \sqrt{n - 1}$ gives $f(x_n) = 1/n \to 0$ , so $0$ is a limit point of the image that the image does not contain. Closedness is therefore not preserved.
5Identify the missing hypothesis. $\mathbb{R}$ is closed but unbounded, so by Heine–Borel it is not compact — the theorem about compactness never applied here.
6Confirm the contrast on a compact domain: on $[-1, 1]$ the same $f$ has image $[\tfrac{1}{2}, 1]$ , since $f(\pm 1) = \tfrac{1}{2}$ and $f(0) = 1$ . That image is closed and bounded, hence compact, exactly as the preservation theorem promises — and the maximum and minimum are attained, which is the extreme value theorem.

No. $f(x) = 1/(1+x^2)$ maps the closed set $\mathbb{R}$ onto $(0,1]$ , which is not closed. Compactness, not closedness, is the property a continuous map carries forward.

Common traps

Treating 'not open' as 'closed'. $[0,1)$ is neither, and $\emptyset$ and the whole space are both.
Assuming closed and bounded implies compact in every metric space. Give an infinite set the discrete metric: it is closed and bounded, but the cover by singletons has no finite subcover.
Assuming continuous maps send open sets to open sets. $x \mapsto x^2$ sends $(-1,1)$ to $[0,1)$ ; continuity constrains preimages only.
Assuming a continuous bijection is a homeomorphism. $t \mapsto (\cos t, \sin t)$ on $[0, 2\pi)$ is a continuous bijection onto the circle with a discontinuous inverse.
Taking arbitrary intersections of open sets to be open. $\bigcap_n (-1/n, 1/n) = \{0\}$ .
Assuming the closure of an open ball is the closed ball of the same radius. In the discrete metric, $B(x,1) = \{x\}$ is already closed, while $\{y : d(x,y) \le 1\}$ is the whole space.
Assuming connected implies path-connected. The topologist's sine curve is the standard counterexample.
Assuming a subspace inherits every property. Compactness and connectedness pass to a subspace only under specific conditions — closed subsets of compact spaces, for instance.

Testing a Planar Set for Openness, Compactness, and Connectedness Metric Spaces and Compactness GRE Mathematics overview Real Analysis Domain Guide Limits and Continuity Sign in for current practice

Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

GRE Subject Test Content and Structure — ETS. Accessed 2026-07-06. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
Mathematical Analysis (Zakon), Section 3.7: Metric Spaces — LibreTexts Mathematics. Accessed 2026-08-15. Zakon's Mathematical Analysis on LibreTexts is CC BY 3.0; attribute the author and platform and avoid verbatim reuse beyond short cited references.
Mathematical Analysis (Zakon), Section 3.8: Open and Closed Sets. Neighborhoods — LibreTexts Mathematics. Accessed 2026-08-15. Zakon's Mathematical Analysis on LibreTexts is CC BY 3.0; attribute the author and platform and avoid verbatim reuse beyond short cited references.
Mathematical Analysis (Zakon), Section 4.6: Compact Sets — LibreTexts Mathematics. Accessed 2026-08-15. Zakon's Mathematical Analysis on LibreTexts is CC BY 3.0; attribute the author and platform and avoid verbatim reuse beyond short cited references.
Mathematical Analysis (Zakon), Section 4.7: More on Compactness — LibreTexts Mathematics. Accessed 2026-08-15. Zakon's Mathematical Analysis on LibreTexts is CC BY 3.0; attribute the author and platform and avoid verbatim reuse beyond short cited references.
Mathematical Analysis (Zakon), Section 4.8: Continuity on Compact Sets. Uniform Continuity — LibreTexts Mathematics. Accessed 2026-08-15. Zakon's Mathematical Analysis on LibreTexts is CC BY 3.0; attribute the author and platform and avoid verbatim reuse beyond short cited references.
Mathematical Analysis (Zakon), Section 4.10: Arcs and Curves. Connected Sets — LibreTexts Mathematics. Accessed 2026-08-15. Zakon's Mathematical Analysis on LibreTexts is CC BY 3.0; attribute the author and platform and avoid verbatim reuse beyond short cited references.

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Topology Domain Guide

Concise answer

Definitions

Intuition

Concept walkthrough

After this page, you should be able to

Formulas and assumptions

Topology axioms

Continuity by preimages

Heine–Borel and its limits

What continuous maps preserve

Compactness and closedness interact

Connectedness in the real line

Worked example

Compactness is preserved; closedness is not

Common traps

Related pages and practice

Sources

Sources and corrections