# Tagged Questions

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### Is the category of these particularly nice spaces cartesian closed?

Is the category of Hausdorff, compactly generated, locally path-connected, semi-locally 1-connected spaces (and continuous maps between them) cartesian closed? If not, in what ways does it fail to be? ...
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### Constructing a Set with Connected Interior

Suppose that $K\subset\mathbb C$ is a compact set with non-empty interior and suppose that $a\in\operatorname{int} K$. I want to construct a set $M$ with the following properties: $M\subseteq K$; ...
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### Compactness and connectedness of the topological space?

Let $X=\mathbb N$ be equipped with the topology generated by the basis consisting of sets $A_n = \{n,n+1,n+2,\ldots\} ,n \in \mathbb N$ . Then $X$ is compact and connected Hausdorff and connected ...
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### Connected and Compact preserving function is not continuous example?

Before we start, I'm aware the result is true for when the function is a map between Euclidean spaces. In fact, with a minimal amount of extra work we can see that a function between locally-compact, ...
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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 ...