Let $X$ be a metric space ; then which of the following is possible ? 1) $X$ has exactly $3$ dense subsets 2) $X$ has exactly $4$ dense subsets 3) $X$ has exactly $5$ dense subsets 4) $X$ has exactly $6$ dense subsets I know that if $X$ has a proper dense subset then for […]

If $(X,\mathcal{T})$ is a connected space, and $Y$ a connected subset, and $X\setminus Y=A\cup B$ for separated sets $A$ and $B$, then why is $A\cup Y$ connected as well? Thank you kindly.

Homework problem, intro to topology. Here’s what I’ve done so far. Am I on the right track? And, how would you advise me to proceed from here? I have already established that $\left |[0,1] \right | = \left |[0,1) \right|$. Now I wish to show that $\left |[0,1) \right| = \left |\mathbb{R} \right |$. I […]

I need represent [0,1] as the union of $2^{\aleph_0}$ perfect sets which are pairwise disjoint. I have thought about removing open disjoint sets but the number of open sets I get is countable. Thanks.

If $f:X \to X$ (codomain and domain have the same topology) is a continuous bijection and every point has finite orbit, is $f^{-1}$ continuous? Note that the orbit being finite and $f$ being a bijection means for all $x$ means for all $x$ there is an $n>0$ such that $f^n(x)=x$. I asked myself this question […]

A prelim problem asked to prove that if $X$ is a compact metric space, and $f:X \to X$ is an isometry (distance-preserving map) then $f$ is surjective. The official proof given used sequences/convergent subsequences and didn’t appeal to my intuition. When I saw the problem, my immediate instinct was that an isometry should be “volume-preserving” […]

Let $f:\mathbb{R}\times[0,1]\to\mathbb{R}$ continuous and $c:\mathbb{R}\to[0,1]$ continuous. Consider $$F:\mathbb{R}\to\mathbb{R},\ \ F(x)=\max_{t\in[0,c(x)]}f(x,t)$$ Is $F$ continuous? I believe it is true, but I’ve difficulties to prove it. I managed to prove that fixing the parameter in one of the two places then the obtained function is continuous, i.e. $$x\mapsto\max_{t\in[0,c(x_0)]}f(x,t) \qquad\text{and}\qquad x\mapsto\max_{t\in[0,c(x)]}f(x_0,t) \qquad\text{are continuous.}$$ But now it seems not […]

Prove that $$ f: \prod\limits_{1}^{\infty} ( \{0,2 \}, \mathcal{T} _{\delta}) \to ([0,1], \mathcal{T}_{e}):\{n_i \} \mapsto \sum_{i=1}^{\infty} \frac{n_i}{3^i} $$ is homeomorphism, and image of $f$ is Cantor set.

Let $E \subset \mathbb{C}$ be compact and totally disconnected. Is there an elementary way to prove that $\mathbb{C} \setminus E$ is connected?

I was wondering which fields $K$ can be equipped with a topology such that a function $f:K \to K$ is continuous if and only if it is a polynomial function $f(x)=a_nx^n+\cdots+a_0$. Obviously, the finite fields with the discrete topology have this property, since every function $f:\Bbb F_q \to \Bbb F_q$ can be written as a […]

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