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1. Let $X$ be a finite set of $n (>1)$ elements and $\tau$ be a topology on $X$ having exactly $m$ elements. Can we give any description of $m$?

2. Does "Invariance of domain" true for injective Darboux function (instead of continuous injection)?

3. How much information can we extract about such a group?

4. If each subring of the fraction field of an integral domain $\mathcal{R}$ (and properly containing $\mathcal{R}$) is a PID, is $\mathcal{R}$ a PID?

5. What condition(s) on $X$ and $A$ can ensure the existence of an element $a\in A$ such that $d(x_0, a)=d(x_0, A) $?

6. Conjecture: Any two matrices of size $n×n$ with same characteristic and minimal polynomial are similar implies $n\le 3$

7. Spaces where each compact subset has compact closure: have they already been studied?

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George Polya said, “If you can’t solve a problem, then there is an easier problem you can’t solve. Find it.”

Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop questioning). – Albert Einstein (1879–1955)

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How Does One Do Mathematical Research?(Or Maybe How Not To)by Lee Lady