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CMI 2025 | Mathematics | Part B | Question: 20
It is known that there exist surjective continuous maps $I \longrightarrow I^{2}$ where $I=[0,1]$ is the unit interval.Using the above result or otherwise, show that ther...
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211
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TIFR Mathematics 2025 | Part A | Question: 3
Consider the set\[S=\left\{(x, y) \in \mathbb{R}^{2} \mid\left(x^{3}+1\right)^{3}=\left(y^{5}+1\right)^{5}\right\} .\]How many connected components does $S$ have?$1$$3$Gr...
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189
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TIFR Mathematics 2025 | Part A | Question: 12
Consider the following subset of $\mathbb{R}^{2}$ :\[S=\left\{\left.\left(x, \sin \frac{1}{x}\right) \right\rvert\, x>0\right\} \cup\left\{\left.\left(0, \pm \frac{1}{n}\...
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161
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TIFR Mathematics 2025 | Part B | Question: 4
There exists an open subset of $\mathbb{R}$ with uncountably many connected components.
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TIFR Mathematics 2025 | Part B | Question: 5
Let $(X, d)$ be a nonempty complete metric space, and let $T: X \rightarrow X$ be a continuous map such that $T^{2}$ is a contraction, i.e., there exists $a<1$ such that ...
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TIFR Mathematics 2025 | Part B | Question: 6
Let\[S^{2}=\left\{(x, y, z) \in \mathbb{R}^{3} \mid x^{2}+y^{2}+z^{2}=1\right\},\]and let $d: S^{2} \times S^{2} \rightarrow \mathbb{R}$ be the restriction of the Euclide...
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TIFR Mathematics 2025 | Part B | Question: 18
There are uncountably many continuous functions $\mathbb{Q} \rightarrow\{0,1\}$.
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TIFR Mathematics 2025 | Part B | Question: 20
If $f, g:[0,1] \rightarrow \mathbb{R}$ are continuous and $f \cdot g=0$, then one of $f$ or $g$ is zero on an open subset of $[0,1]$.
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137
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CMI 2024 | Mathematics | Part B | Question: 2
Let $\mathrm{SL}(2, \mathbb{R})$ be the group of $2 \times 2$ matrices with real entries and determinant 1 , endowed with the subspace topology of $\mathbb{R}^{4}$. Consi...
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CMI 2024 | Mathematics | Part B | Question: 9
Let $M_{n}(\mathbb{R})$ be the space of $n \times n$ matrices with real entries, identified with the euclidean space $\mathbb{R}^{n^{2}}$. Let $X$ be a compact subset of ...
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CMI 2024 | Mathematics | Part A | Question: 6
Let $X$ be a subset of $\mathbb{R}^{3}$. We say that $X$ has property $S$ if it contains at least two elements and every Cauchy sequence in $X$ has a limit point in $X$. ...
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CMI 2024 | Mathematics | Part A | Question: 10
Let $X=\{\alpha \in \mathbb{C} \mid \alpha$ satisfies a monic polynomial over $\mathbb{Q}\}$. (I.e., $X$ is the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$.) Endow ...
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CMI 2023 | Mathematics | Part A | Question: 3
Let $X$ be a compact topological space. Let $f: X \longrightarrow \mathbb{R}$ be a function satisfying $f^{-1}([n, \infty))$ is closed for all $n \in \mathbb{N}$. Pick th...
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TIFR Mathematics 2024 | Part B | Question: 2
There exists a metric space $\text{X}$ such that the number of open subsets of $\text{X}$ is exactly $2024$.
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TIFR Mathematics 2024 | Part B | Question: 6
The complement of any countable union of lines in $\mathbb{R}^{3}$ is path connected.
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TIFR Mathematics 2024 | Part B | Question: 7
The subsets $\left\{(x, y) \in \mathbb{R}^{2} \mid\left(y^{2}-x\right)\left(y^{2}-x-1\right)=0\right\}$ and $\left\{(x, y) \in \mathbb{R}^{2} \mid y^{2}-x^{2}=1\right\}$ ...
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TIFR Mathematics 2023 | Part B | Question: 11
Answer whether the following statements are True or False.There exists a continuous bijection from $[0,1] \times[0,1]$ to $\left\{(x, y) \in \mathbb{R}^{2} \mid x^{2}+y^{...
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TIFR Mathematics 2023 | Part B | Question: 12
Answer whether the following statements are True or False.Let $f \in \mathbb{C}\left[z_{1}, \ldots, z_{n}\right]$ be a nonzero polynomial $(n \geq 1),$ and let\[X=\left\{...
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TIFR Mathematics 2022 | Part B | Question: 1
Answer whether the following statements are True or False.$\mathbb{R}^{2} \backslash \mathbb{Q}^{2}$ is connected but not path-connected.
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TIFR Mathematics 2022 | Part B | Question: 5
Answer whether the following statements are True or False.Let $A \subset \mathbb{R}^{2}$ be a nonempty subset such that any continuous function $f: A \rightarrow \mathbb{...
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TIFR2021-Maths-B: 3
Every infinite closed subset of $\mathbb{R}^n$ is the closure of a countable set.
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TIFR-2016-Maths-B: 25
Let $S^{1}= \left \{ z \in \mathbb{C} :\left | z \right | =1\right \}$ be the unit circle. Which of the following is false ? Any continuous function from $S^{1}$ to $\mat...
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TIFR-2016-Maths-B: 30
For $X\subset \mathbb{R}^{n}$, consider $X$ as a metric space with metric induced by the usual Euclidean metric on $\mathbb{R}^{n}$. Which of the following metric spaces ...
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TIFR-2017-Maths-A: 15
True/False Question :Under the projection map $\mathbb{R}^{2}\rightarrow \mathbb{R}$ sending $\left ( x,y \right )$ to $x$, the image of any closed set is closed.
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TIFR-2017-Maths-A: 21
True/False Question :There is a continuous onto function $f:S^{2}\rightarrow S^{1}$ from the unit sphere in $\mathbb{R}^{3}$ to the unit sphere in $\mathbb{R}^{2}$, where...
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TIFR-2018-Maths-A: 22
True/False Question :There exists a continuous surjection from $\mathbb{R}^{3}-S^{2}$ to $\mathbb{R}^{2}-\left \{ \left ( 0,0 \right ) \right \}$ (here $S^{2}\subset \mat...
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343
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TIFR-2019-Maths-B: 11
True/False Question :If $0$ is a limit point of a set $A\subseteq \left ( 0,\infty \right )$, then the set of all $x\in\left ( 0,\infty \right )$ that can be expressed a...
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TIFR-2019-Maths-A: 17
Let $X\subset \mathbb{R}^{2}$ be the subset$$X=\left \{ \left ( x,y \right ) \left | x=0, \right |y \mid \leq 1\right \}\cup \left \{ \left ( x,y \right ) \mid 0 < x \leq...
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349
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TIFR-2020-Maths-B: 5
True/False Question :If $X$ is a metric space such that every continuous function $f:X\rightarrow \mathbb{R}$ is uniformly continuous, then $X$ is compact.
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343
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TIFR-2020-Maths-B: 7
True/False Question :Let $A$ be a countable union of lines in $\mathbb{R}^{3}$. Then $\mathbb{R}^{3} \setminus A$ is connected.
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