# Tagged Questions

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; ...

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### Give example of $f$ that is open but neither closed not continuous (in 2D).

I'm trying to teach my self topology. The book I'm using has the following problem: Give an example of two subsets $X,Y \subseteq \mathbb R ^2$, both considered as topological spaces with their ...
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### Example of Kolmogorov space which is neither Fréchet nor sober

Is every Kolmogorov ($T_0$) space either Fréchet ($T_1$) or sober or both, or is it possible to be Kolmogorov but neither of those two? If the latter, can a counterexample be defined? A sober space ...
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### Prove $(\text L)\int_0^1[x-\text K(x)]\sin x\text d x= (\text L)\int_0^1x\sin x \text dx$

Let $\text K(x)$ be a Cantor function on $[0,1]$ prove $$(\text L)\int_0^1[x-\text K(x)]\sin x\text d x= (\text L)\int_0^1x\sin x \text dx$$ here $(\text L)$ denotes Lebesgue-integral. Attempt: ...
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### Helly theorem application

Let $X$ be a normed space, $dim(X)=d$, let $r>0, A$ and $\subset X$ . Show that if every $d+1$ points of $A$ are contained in a closed ball of radius $r$, then $A$ is contained in a closed ball of ...
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### Box topology definition

Munkres defines the box topology as shown below. I am trying to understand how come an element of basis as defined is even subset of $\prod_{\alpha \in J}X_{\alpha}$. If we get an element of the basis,...
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### Finding closure of image of operator

I'm working on an old exam problem: Define for $u \in C^2([-1,1])$ the operator $L$ by $[Lu](x) = - \frac{d}{dx} \left( (1-x^2) u'(x) \right)$. Set $\Omega = \{ Lu \mid u \in C^2([-1,1]) \}$. Find the ...
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### Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true?

Let $X$ be a non-empty compact Hausdorff space. Which of the following statements are true? $a.$ If $X$ has at least $n$ distinct points, then the dimension of $C(X)$, the space of continuous real ...
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### Is there a topology which is coarser than the product topology?

Is there a topology which is coarser than the product topology on an infinite Cartesian product of topological spaces?
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### To find two points in compact metric space satisfying specific property

Let $(X,d)$ be a compact metric space.Suppose that for all positive real numbers $t<1$ ,there are points $x_{t}$,$y_{t}$ such that $d(x_{t},y_{t})=t$.Prove that there are points $x$, $y$ in $X$ ...
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### Product of perfectly-$T_3$ spaces

This is (sort of) a continuation of this question Let $X$ be a topological space. Then, we say that $X$ is perfectly-$T_3$ iff it is $T_1$ and for every $x\in X$ and closed subset $C\subseteq X$ not ...
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### what is the nature of a ball that goes over a “corner” of the real projective plane?

I'm make a little computer program to help me understand different 2d topological spaces, (such as torus and mobius band). I'm having issues with drawing balls that go over a corner of the real ...
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### Three Jordan curves made from paths

For continuous paths $\phi,\psi:[0,1]\to \mathbb{R}^2$ such that $\phi(1)=\psi(0)$, let $\phi*\psi$ denote the composition path and $\phi^{-1}$ the inverse path. Consider points $p,q\in\mathbb{R}^2$ ...
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### Surjectivity on the image of a annulus

I'm trying to prove the Fundamental Theorem of Algebra as it is done in Birkhoff and MacLane. Unfortunately, I don't have access to the book, only to a sketch. Therefore, I'm filling the gaps myself. ...
Let $X$ be a complete separable metric space containing no perfect set of size greater than $1$. In other words every subset of $X$ has an isolated point. It is well known that $X$ must be countable....