List questions from examples-counterexamples

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Generate a simple counterexample

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If every bijective map $M\to M$ is a homeomorphism, when is $(M, \tau)$ discrete?

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Product of open and closed set is open or closed?

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Does basis of $A$ in $4$ dimensions, make basis for $B$

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Find infinitely many homomorphism from $GL_2(\mathbb Q)$ to $\mathbb Q^*$

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Is there $f \in L^p(\Omega)$ such that given $\epsilon>0$, $f \not \in L^{p+\epsilon}(\Omega)$?

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If $f$ preserves identity and inverses, is it necessarily a group homomorphism?

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Examples of non abelian powerful p-groups

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List Question

Given an example in NLS $X$ to show that if $A,F$ is closed then $A+F$ is not closed

Generate a simple counterexample

If every bijective map $M\to M$ is a homeomorphism, when is $(M, \tau)$ discrete?

Product of open and closed set is open or closed?

Counterexample to Liouville in reals

Does basis of $A$ in $4$ dimensions, make basis for $B$

Example of topological space that is connected, locally connected but not path connected

Almost sure convergence of rescaled nondecreasing sequences of random variables

Find infinitely many homomorphism from $GL_2(\mathbb Q)$ to $\mathbb Q^*$

Is there $f \in L^p(\Omega)$ such that given $\epsilon>0$, $f \not \in L^{p+\epsilon}(\Omega)$?

Restriction on the injective, continuous image of $\mathbb{R}$ into $\mathbb{R}^2$.

A cardinal relation on Banach spaces,

If $f$ preserves identity and inverses, is it necessarily a group homomorphism?

Examples of non abelian powerful p-groups

Topological "Freshman's Dream"

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