# Tagged Questions

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### Show that f is onto.

Let $X$ be a compact connected Hausdorff space and $f:X\rightarrow X$ a continuous open map. Show that f is onto.
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### A question of topology.

If S is a subset of $\hspace{0.1cm}$$[0,1]\times[0,1]$$\hspace{0.1cm}$ such taht one point of the ordered pair is rational and the other is irrational or both are irrationals,then which of the ...
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### How to show $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ is path connected and compact?

let $A=\{(x,y)\in R^2:4x^2+9y^2=36\}$ . Show that A is path connected and compact. my attempt: since $\frac {x^2}{9}+\frac{y^2}{4}=1$ is elips. A is bounded and closed. so is compact. (by heine ...
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### Constant Function over Connected, Compact Space

I am working on this problem and was wondering if I could get some feedback on my attempt at the proof. My gut tells me that I need a stronger argument as why my covering is actually a cover. I also ...
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### Any ball is connected?

Let $X$ be a compact , metric space. Assume that the closure of every each open ball it the closed ball with same center and radius. Prove that any ball in this space is connected.
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### Example of topological space where pseudo-component differ with intersection of clopen sets.

It is well known fact that connected component $C_x$ of a point $x$ from some topological space $\tau$ is contained in every clopen set containing $x$ (so it's intersection $M$ also contains $C_x$). ...
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### Proofs that quasicomponents of compact Hausdorff spaces are connected

Nuno's answer to Any two points in a Stone space can be disconnected by clopen sets uses (and proves) the following: Theorem: Let $X$ be a compact Hausdorff space. Then the quasicomponents of $X$ are ...
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### Long line is connected and compact

How to prove that the long line is connected and compact. I was trying to prove connectedness using contradiction but couldn't.
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### What about the compactness and connectedness of $[0,1]$ in this topology?

If the set $\mathbf{R}$ of all real numbers has the topology consisting of all sets $A$ such that $\mathbf{R} \setminus A$ is either countable or all of $\mathbf{R}$. What can we say about compactness ...
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### Prove that $\bigcap_{k = 1}^\infty C_k$ is also compact and connected. [duplicate]

Let $X$ be a Hausdorff space and let $C_0 \supset C_1 \supset ...$ be a decreasing sequence of compact connected subsets of $X$. Prove that $\bigcap_{k = 1}^\infty C_k$ is also compact and connected. ...
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### Is this set compact? Connected?

Is this set compact? Connected? $S=\{(x,y,z)\in\mathbb{R}^3:z=x^2+y^2+1\}$ for $z\le 1$ this set is not defined, but for $z>1$ we are getting circles! I imagined this as a bunch of ...
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### Show that $X$ is locally compact, and each connected component of $X$ is a point.

Let $X$ be a Hausdorff space and suppose that each point has a neighborhood basis of simultaneously open and compact neighborhoods. Show that $X$ is locally compact, and each connected component of ...
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### Show $X=\left\{x \in [0,1]: x \neq \frac1n\text{ for any }n \in \Bbb N\right\}$ is neither compact nor connected

I am stuck on the following question: Let $X=\{x \in [0,1]: x \neq \frac1n: n \in \Bbb N\}$ be given the subspace topology. Then I have to prove that $X$ is neither compact nor connected. Can ...
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### Prove that $\bigcap _{i=1}X_i$ is connected. [duplicate]

Let $X$ be a compact Hausdorff space and let $X_1 \subset X_2 \subset X_3 \subset \cdots$ be a sequence of closed, connected subspaces. Prove that $\bigcap_{i=1}^\infty X_i$ is connected. Give an ...
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Call an open cover $\mathscr{U}$ of a metric space $M$ strongly additive if whenever $U,V\in\mathscr{U}$ and $U\cap V\ne\emptyset$, then $U\cup V\in\mathscr{U}$. Prove that $M$ is compact and ...
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### Locally connected and compact Hausdorff space invariant of continuous mappings

I am looking for a reference (not proof) to the following theorem: If $X$ is a compact and locally connected topological space, Y is a Hausdorff topological space, $f:X\to Y$ is continuous and ...
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### A restricted continuous map is a homeomorphism

Suppose that $f:M\rightarrow N$ is a continuous map with the property that $\forall x\in M\exists$ open neighbourhood $U\subset M$ with $x\in U$ and open neighbourhood $V\subset N$ with $f(x)\in V$ ...
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### Compactness and connectedness on $M_n(\mathbb R)$

Consider $M_n(\mathbb R)$, the set of all $n\times n$ matrices. Which of the following are compact and which are connected? a) The set of all invertible matrices b) The set of all orthogonal ...