List of top Mathematics Questions on Topology

Let \( C[0,1] = \{ f: [0,1] \to \mathbb{R} : f \text{ is continuous}\} \) and \[ d_\infty(f,g) = \sup\{ |f(x) - g(x)|: x \in [0, 1]\} \] for \( f, g \in C[0,1] \).
For each \( n \in \mathbb{N} \), define \( f_n: [0,1] \to \mathbb{R} \) by \( f_n(x) = x^n \) for all \( x \in [0,1] \).
Let \( P = \{f_n: n \in \mathbb{N}\} \).
Which of the following statements is/are correct?

Let X and Y be two topological spaces. A continuous map \(f: X \to Y\) is said to be proper if \(f^{-1}(K)\) is compact in X for every compact subset K of Y, where \(f^{-1}(K)\) is defined by \(f^{-1}(K) = \{x \in X : f(x) \in K\}\).
Consider \(\mathbb{R}\) with the usual topology. If \(\mathbb{R} \setminus \{0\}\) has the subspace topology induced from \(\mathbb{R}\) and \(\mathbb{R} \times \mathbb{R}\) has the product topology, then which of the following maps is proper?

Consider the metrics \(\rho_1\) and \(\rho_2\) on \(\mathbb{R}\), defined by
\[ \rho_1(x, y) = |x-y| \quad \text{and} \quad \rho_2(x, y) = \begin{cases} 0, & \text{if } x = y \\ 1, & \text{if } x \neq y \end{cases} \]
Let \(X = \{n \in \mathbb{N} : n \ge 3\}\) and \(Y = \{n + \frac{1}{n} : n \in \mathbb{N}\}\).
Define \(f: X \cup Y \to \mathbb{R}\) by \[ f(x) = \begin{cases} 2, & \text{if } x \in X \\ 3, & \text{if } x \in Y \end{cases} \]
Consider the following statements:
P: The function \(f: (X \cup Y, \rho_1) \to (\mathbb{R}, \rho_1)\) is uniformly continuous.
Q: The function \(f: (X \cup Y, \rho_2) \to (\mathbb{R}, \rho_1)\) is uniformly continuous.
Then

Let \(C[0, 1] = \{ f : [0, 1] \to \mathbb{R} : f \text{ is continuous}\}\).
Consider the metric space \((C[0,1], d_\infty)\), where \[ d_\infty(f, g) = \sup\{ |f(x) - g(x)| : x \in [0, 1] \} \text{ for } f, g \in C[0,1]. \] Let \(f_0(x) = 0\) for all \(x \in [0,1]\) and \[ X = \{f \in (C[0, 1], d_\infty) : d_\infty(f_0, f) \ge \frac{1}{2}\}. \] Let \(f_1, f_2 \in C[0, 1]\) be defined by \(f_1(x) = x\) and \(f_2(x) = 1-x\) for all \(x \in [0,1]\).
Consider the following statements:
P: \(f_1\) is in the interior of X.
Q: \(f_2\) is in the interior of X.
Which of the following statements is correct?

Consider \(\mathbb{R}^2\) with the usual Euclidean metric. Let
\[ X = \left\{\left(x, x \sin\frac{1}{x}\right) \in \mathbb{R}^2 : x \in (0,1]\right\} \cup \{(0, y) \in \mathbb{R}^2 : -\infty < y < \infty\}, \]
\[ Y = \left\{\left(x, \sin\frac{1}{x}\right) \in \mathbb{R}^2 : x \in (0,1]\right\} \cup \{(0, y) \in \mathbb{R}^2 : -\infty < y < \infty\}. \]

Consider the following statements:
P: \(X\) is a connected subset of \(\mathbb{R}^2\).
Q: \(Y\) is a connected subset of \(\mathbb{R}^2\).
Then

Let \(\mathbb{R}^3\) be a topological space with the usual topology and \(\mathbb{Q}\) denote the set of rational numbers. Define the subspaces X, Y, Z and W of \(\mathbb{R}^3\) as follows:
\(X = \{(x, y, z) \in \mathbb{R}^3 : |x| + |y| + |z| \in \mathbb{Q}\}\)
\(Y = \{(x, y, z) \in \mathbb{R}^3 : xyz = 1\}\)
\(Z = \{(x, y, z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1\}\)
\(W = \{(x, y, z) \in \mathbb{R}^3 : xyz = 0\}\)
Which of the following statements is correct?

Let \((\mathbb{R}, \tau)\) be a topological space, where the topology \(\tau\) is defined as \[ \tau = \{U \subseteq \mathbb{R} : U = \emptyset \text{ or } 1 \in U\}. \] Which of the following statements is/are correct?

Consider the intervals \( S = (0, 2] \) and \( T = [1, 3] \). Let \( S^\circ \) and \( T^\circ \) be the sets of interior points of \( S \) and \( T \), respectively. Then the set of interior points of \( S \setminus T \) is equal to

The set \[ \left\{ \frac{x}{1+x^2} : -1<x<1 \right\}, \text{ as a subset of } \mathbb{R}, \text{ is} \]

The set \[ \left\{ \frac{1}{m} + \frac{1}{n} : m, n \in \mathbb{N} \right\} \cup \{0\}, \text{ as a subset of } \mathbb{R}, \text{ is} \]