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The standard topology and the lower limit topology on $\mathbb{R}$ have the same dense subsets. They are two different topologies(even up to homeomorphism) on the real line. So the next question c …

Assume that $f,g:S^{2n}\to \mathbb{R}^{n}$ are $2$ maps. Assume that the set valued map $p(x)=\{f(x),g(x)\}$ is a continuous set valued map. Does there exist a point $p\in S^{2n}$ such that $p(x)=p( …

Let $X$ be a compact Haussdorf topological space with the following property: For every continuous function $f:X\to \mathbb{C}$ the image $f(X)$ is a path connected subset of $\mathbb{C}$. Is suc …

Let $(X,T)$ be a compact metrizable space. For every metric $d$ on $X$ which is $T-$ compatible, the Hausdorff metric on $2^{X}$ gives a topology on $2^{X}$. (Up to homeomorphism) is this topology i …

We identify the vector space tensor product $\mathbb{R}^{m} \otimes \mathbb{R}^{n}$ with $\mathbb{R}^{mn}$ Let $X$ be the space of all non zero simple tensors $X=\{a\otimes b \mid a\in \math …

Is there a category $\mathcal{A}$ of topological spaces, saturated with respect to homeomorphism relation, which is maximal with respect to the following property? For every $X,Y \in \mathcal{A …

Are there two non homeomorphic continua $X,Y$ such that $X $ can be embedded in $Y$ but there is no topological space $Z$ with $$X<Z<Y.$$ The later relation means that $Z$ is h …

Let $X$ be the subset of the long line consist of rational points with the topology inherits from the long line. Is $X$ a metrizable space?

Motivated by this question we ask: Up to homeomorphism, are there only a finite number of connected locally compact hausdorff topological space $X$ such that $X$ has an open set $U$ homeomorp …

Let $X$ be a compact metric space. A $k$ mean on $X$ is a continuous map $f:X^{k}\to X$ which is identity on the diagonal and is invariant under all $k$-permutations. For details, See the following …

I asked this question at MSE but I did not received any answer, so I repeat it here at MO: What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X) …

Is there a continuous surjective function $f:[0,1] \to [0,1]^2$ such that every level set $f^{-1}(y)$ is a finite set? If the answer is no, what about if we replace the finiteness of level sets by "co …

Is there a topological space $X$, which is not a singleton, and satisfies the following property? For every continuous function $f: X\times S^2\to\mathbb{R}^2$ there exist a point $x\in S^2$ such th …

Put $P=[0,1]$. Is there a compact subset $L$ of the hyper space of $P$ such that the pair $(P, L)$ satisfies the following axioms of projective geometry. Furthermore the obvious maps from the config …

Is it true to say that a compact hausdorff space $X$ is path connected if and only if for every continuous function $f:X\to \mathbb{C}$, we have $f(X)\subset \mathbb{C}$ is path connected? …