GATE Mathematics Topology Syllabus

Define compactness of a topological space $X$.

$X$ is compact if every open cover of $X$ has a finite subcover: whenever $X=\bigcup_{\alpha} U_\alpha$ with each $U_\alpha$ open, there is a finite subset such that $X=U_{\alpha_1}\cup\cdots\cup U_{\alpha_n}$.

State the Heine–Borel Theorem characterizing compact subsets of $\mathbb{R}^n$.

A subset of $\mathbb{R}^n$ (Euclidean topology) is compact iff it is closed and bounded.

What does the continuous image of a compact space satisfy, and state the related extreme value result.

If $f:X\to Y$ is continuous and $X$ is compact, then $f(X)$ is compact. Consequently a continuous $f:X\to\mathbb{R}$ on a compact $X$ is bounded and attains its maximum and minimum (Extreme Value Theorem).

State Tychonoff's Theorem.

An arbitrary product $\prod_{\alpha\in J} X_\alpha$ of compact spaces is compact in the product topology.

In a Hausdorff space, what can be said about compact subsets, and what about closed subsets of a compact space?

In a Hausdorff space every compact subset is closed. In any compact space every closed subset is compact.

State the theorem about a continuous bijection from a compact space to a Hausdorff space.

A continuous bijection $f:X\to Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism (it is automatically a closed map, hence its inverse is continuous).

Define the $T_1$ separation axiom.

A space is $T_1$ if for every pair of distinct points $x\neq y$ there is an open set containing $x$ but not $y$ (and one containing $y$ but not $x$). Equivalently, every singleton $\{x\}$ is closed.

Define the $T_2$ (Hausdorff) separation axiom.

A space $X$ is Hausdorff if for every pair of distinct points $x\neq y$ there exist disjoint open sets $U\ni x$ and $V\ni y$.

Define regular ($T_3$) and normal ($T_4$) spaces.

Regular ($T_3$): $T_1$, and for each point $x$ and closed set $C$ with $x\notin C$ there are disjoint open sets separating them. Normal ($T_4$): $T_1$, and any two disjoint closed sets can be separated by disjoint open sets.

State the chain of implications among the separation axioms $T_1$ through $T_4$.

$T_4\Rightarrow T_3\Rightarrow T_2\Rightarrow T_1$ (using the $T_1$ assumption built into regular/normal). None of the reverse implications hold in general.

Define first countability and second countability.

First countable: every point has a countable neighborhood basis (local base). Second countable: the whole topology has a countable basis. Second countable $\Rightarrow$ first countable.

Define a separable space and a Lindelöf space.

Separable: the space has a countable dense subset. Lindelöf: every open cover has a countable subcover. Second countability implies both separability and the Lindelöf property.

In a metrizable space, which of second countable, separable, and Lindelöf are equivalent?

For metrizable spaces the three properties second countable, separable, and Lindelöf are all equivalent.

State the relationship between compactness and the Lindelöf property.

Compactness (every open cover has a finite subcover) is strictly stronger than Lindelöf (every open cover has a countable subcover); every compact space is Lindelöf, but not conversely (e.g. $\mathbb{R}$ is Lindelöf, not compact).

State Urysohn's Lemma.

A space $X$ is normal iff 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 $f(B)=\{1\}$.

In Urysohn's Lemma, do the sets $A$ and $B$ have to be exactly the preimages $f^{-1}(0)$ and $f^{-1}(1)$?

No. The lemma only guarantees $f(A)=\{0\}$ and $f(B)=\{1\}$; in general $A\subseteq f^{-1}(0)$ and $B\subseteq f^{-1}(1)$ and these inclusions may be proper. The strengthening to equality requires perfect normality ($G_\delta$ closed sets).

State the Urysohn Metrization Theorem.

Every regular ($T_3$) space with a countable basis (second countable) is metrizable. Equivalently, a second countable space is metrizable iff it is regular (Hausdorff).

State the Tietze Extension Theorem and its connection to normality.

If $X$ is normal and $A\subseteq X$ is closed, then any continuous $f:A\to\mathbb{R}$ (or into $[a,b]$) extends to a continuous function on all of $X$. It is equivalent to normality and is proved using Urysohn's Lemma.

Define the closure $\overline{A}$ of a subset $A$ in terms of limit points and in terms of closed sets.

$\overline{A}$ equals $A$ together with all its limit points; equivalently it is the smallest closed set containing $A$, i.e. the intersection of all closed sets that contain $A$.

State the closure criterion: when is $x\in\overline{A}$?

$x\in\overline{A}$ iff every open set (equivalently every basis element) containing $x$ intersects $A$. In a first countable space this is equivalent to $x$ being the limit of a sequence in $A$.

Define a continuous map $f:X\to Y$ using preimages of open sets, and give an equivalent closed-set form.

$f$ is continuous if $f^{-1}(U)$ is open in $X$ for every open $U\subseteq Y$. Equivalently, $f^{-1}(C)$ is closed for every closed $C\subseteq Y$; equivalently $f(\overline{A})\subseteq\overline{f(A)}$ for all $A\subseteq X$.

Define a homeomorphism between topological spaces.

A bijection $f:X\to Y$ such that both $f$ and $f^{-1}$ are continuous; equivalently a continuous bijection that is also an open (equivalently closed) map. Spaces related by one are topologically indistinguishable.

Compare an open map, a closed map, and a continuous map.

Continuous: preimages of open sets are open. Open map: images of open sets are open. Closed map: images of closed sets are closed. These three notions are independent; a homeomorphism is all three (plus bijective).

State the Lebesgue Number Lemma.

If $X$ is a compact metric space and $\mathcal{A}$ is an open cover of $X$, then there exists $\delta>0$ (a Lebesgue number) such that every subset of $X$ of diameter less than $\delta$ is contained in some member of $\mathcal{A}$.

State the relationship between compactness, limit point compactness, and sequential compactness in general and in metric spaces.

In general: compact $\Rightarrow$ limit point compact, and these are distinct from sequential compactness. In a metrizable space the three notions — compact, limit point compact, and sequentially compact — are all equivalent.

Give two standard metrics inducing the same product topology on $\mathbb{R}^n$ and one that does not (different metric, same topology).

The Euclidean metric $d(x,y)=\sqrt{\sum (x_i-y_i)^2}$, the taxicab metric $\sum|x_i-y_i|$, and the sup metric $\max_i|x_i-y_i|$ all induce the same (standard product) topology on $\mathbb{R}^n$; they are topologically equivalent though numerically different.

State the pasting (gluing) lemma for continuous functions.

If $X=A\cup B$ with $A,B$ both closed (or both open), and $f:A\to Y$, $g:B\to Y$ are continuous with $f=g$ on $A\cap B$, then the combined function $h:X\to Y$ defined by $f$ on $A$ and $g$ on $B$ is continuous.

How are connected components and path components related, and what are their key properties?

Components are the maximal connected subsets; they are closed, disjoint, and partition $X$. Path components are maximal path-connected subsets and partition $X$ too. Each path component lies in a single component; in general components may be larger than (unions of) path components.

Define the quotient space $X/{\sim}$ for an equivalence relation $\sim$ on $X$.

$X/{\sim}$ is the set of equivalence classes with the quotient topology induced by the projection $p:X\to X/{\sim}$ sending each point to its class: $U$ is open iff $p^{-1}(U)$ is open. E.g. $[0,1]$ with $0\sim 1$ gives a circle $S^1$.

Is the product of connected (resp. compact) spaces connected (resp. compact)?

Yes to both: an arbitrary product of connected spaces is connected, and by Tychonoff's Theorem an arbitrary product of compact spaces is compact, both in the product topology.