GATE Mathematics Topology Syllabus

Topology syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Topology in GATE Mathematics, not a summary of it.

1. 1

   Basic concepts of topology

   11 topics

   • Bases
   • Subbases
   • Subspace topology
   • Order topology
   • Product topology
   • Quotient topology
   • Metric topology
   • Connectedness
   • Compactness
   • Countability and separation axioms
   • Urysohn’s Lemma

Topology flashcards for GATE Mathematics

22 of 50 cards from the Topology deck — real questions with worked answers.

1. What is a basis (base) $\mathcal{B}$ for a topology on a set $X$?

   A collection $\mathcal{B}$ of subsets of $X$ such that (1) every $x\in X$ lies in some $B\in\mathcal{B}$, and (2) if $x\in B_1\cap B_2$ with $B_1,B_2\in\mathcal{B}$, then there exists $B_3\in\mathcal{B}$ with $x\in B_3\subseteq B_1\cap B_2$.

2. How is the topology $\tau$ generated by a basis $\mathcal{B}$ defined?

   A set $U\subseteq X$ is open ($U\in\tau$) if and only if for every $x\in U$ there exists $B\in\mathcal{B}$ with $x\in B\subseteq U$. Equivalently, every open set is a union of basis elements.

3. State the criterion (using bases) for a topology $\tau'$ to be finer than a topology $\tau$ on $X$.

   Let $\mathcal{B},\mathcal{B}'$ generate $\tau,\tau'$. Then $\tau\subseteq\tau'$ iff for each $B\in\mathcal{B}$ and each $x\in B$ there is $B'\in\mathcal{B}'$ with $x\in B'\subseteq B$.

4. What is a subbasis $\mathcal{S}$ for a topology on $X$, and how does it generate the topology?

   A subbasis is any collection $\mathcal{S}$ of subsets of $X$ whose union is $X$. The topology it generates has as a basis all finite intersections $S_1\cap\cdots\cap S_n$ of members of $\mathcal{S}$; open sets are arbitrary unions of these.

5. Given a topological space $(X,\tau)$ and $Y\subseteq X$, define the subspace topology on $Y$.

   The subspace (relative) topology is $\tau_Y=\{\,Y\cap U : U\in\tau\,\}$, i.e. open sets of $Y$ are intersections of open sets of $X$ with $Y$.

6. If $\mathcal{B}$ is a basis for $X$ and $Y\subseteq X$, what is a basis for the subspace topology on $Y$?

   The collection $\mathcal{B}_Y=\{\,B\cap Y : B\in\mathcal{B}\,\}$ is a basis for the subspace topology on $Y$.

7. When a set $A\subseteq Y\subseteq X$ is open (or closed) in $Y$, when is it open (or closed) in $X$?

   If $Y$ is open in $X$, then $A$ open in $Y$ implies $A$ open in $X$. If $Y$ is closed in $X$, then $A$ closed in $Y$ implies $A$ closed in $X$. In general a set open/closed in the subspace need not be so in $X$.

8. Define the order topology on a totally ordered set $(X,<)$ with more than one element.

   It is generated by the basis of all open intervals $(a,b)=\{x:a<x<b\}$, together with all rays $[a_0,b)$ if $X$ has a smallest element $a_0$ and all rays $(a,b_0]$ if $X$ has a largest element $b_0$.

9. Define the product topology on $\prod_{\alpha\in J} X_\alpha$ and give its standard subbasis.

   It is generated by the subbasis $\{\pi_\beta^{-1}(U_\beta): \beta\in J,\ U_\beta\subseteq X_\beta \text{ open}\}$. A basis element is $\prod_\alpha U_\alpha$ where $U_\alpha$ is open and $U_\alpha=X_\alpha$ for all but finitely many $\alpha$.

10. How does the box topology on $\prod_\alpha X_\alpha$ differ from the product topology?

    The box topology has basis $\prod_\alpha U_\alpha$ with each $U_\alpha$ open and no finiteness restriction. For finite products box = product; for infinite products the box topology is strictly finer.

11. State the universal mapping property characterizing the product topology.

    A function $f:Z\to\prod_\alpha X_\alpha$ is continuous iff each coordinate function $\pi_\alpha\circ f:Z\to X_\alpha$ is continuous. (This characterizes the product topology, not the box topology.)

12. Define the quotient topology induced by a surjection $p:X\to Y$ from a space $X$.

    A set $U\subseteq Y$ is open iff $p^{-1}(U)$ is open in $X$. This is the finest topology on $Y$ making $p$ continuous; $p$ is then called a quotient map.

13. What is a quotient map $p:X\to Y$ (give the defining condition)?

    A surjective continuous map such that $U\subseteq Y$ is open iff $p^{-1}(U)$ is open in $X$ (equivalently the same for closed sets). Continuous surjections that are open maps, or closed maps, are quotient maps.

14. Define a metric topology on a set $X$ induced by a metric $d$.

    It is generated by the basis of open balls $B_d(x,\varepsilon)=\{y\in X: d(x,y)<\varepsilon\}$ for all $x\in X$ and $\varepsilon>0$. A set is open iff each of its points has an open ball around it contained in the set.

15. State the four axioms a metric $d:X\times X\to\mathbb{R}$ must satisfy.

    For all $x,y,z$: (1) $d(x,y)\geq 0$; (2) $d(x,y)=0 \iff x=y$; (3) symmetry $d(x,y)=d(y,x)$; (4) triangle inequality $d(x,z)\leq d(x,y)+d(y,z)$.

16. What does it mean for a topological space to be metrizable?

    A space $(X,\tau)$ is metrizable if there exists a metric $d$ on $X$ whose induced metric topology equals $\tau$.

17. Define connectedness of a topological space $X$.

    $X$ is connected if it cannot be written as a union $X=A\cup B$ of two disjoint nonempty open sets (a separation). Equivalently, the only subsets that are both open and closed (clopen) are $\varnothing$ and $X$.

18. Characterize the connected subsets of $\mathbb{R}$ (with the standard topology).

    A subset of $\mathbb{R}$ is connected iff it is an interval (including rays, $\mathbb{R}$, single points, and $\varnothing$), i.e. a convex set: if it contains $a<c<b$ with $a,b$ in it, then $c$ is in it.

19. State the theorem on continuous images of connected spaces, and name its classical corollary.

    If $f:X\to Y$ is continuous and $X$ is connected, then $f(X)$ is connected. Corollary: the Intermediate Value Theorem — a continuous $f:[a,b]\to\mathbb{R}$ attains every value between $f(a)$ and $f(b)$.

20. What is the relationship between path-connectedness and connectedness?

    Path-connected $\Rightarrow$ connected, but not conversely. Counterexample: the topologist's sine curve is connected but not path-connected.

21. 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 $X=U_{\alpha_1}\cup\cdots\cup U_{\alpha_n}$.

22. 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.

See more Topology flashcards →

Planning Topology for GATE Mathematics

Topology (GATE Mathematics) FAQ