# Tagged Questions

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### Main Theorems/Techniques for proving Homeomorphism?

General Question: what are the most common Theorems/Methods used to prove Homeomorphism? I encountered: - find the map explicitly - use the Compact-to-Hausdorff Lemma - find cts maps $f$ and $g$ ...
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### Metric-like families of relations

Let $X$ be an arbitrary set and to start with, let us consider a relation $\leq$ on $X$ (that is $\leq$ is a subset of $X^2$) which is reflexive and transitive. such a relation is called a preorder. ...
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### Hemicontinuity of multifunctions/correspondences that can map to the empty set

The Wikipedia article on hemicontinuity of multifunctions or correspondences does not make it clear whether the multifunction or correspondence $f : A \to 2^B$ (power set) is allowed to map to the ...
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### Relation as the Union of 4 Relations

I'm trying to write the relation $$\rho=\{\langle{x},y\rangle\in{\mathbb{R}\times\mathbb{R}}: |x|+2|y|=1\}$$ as the union of 4 relations. Is it enough to just think of this as a diamond and use the ...
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### Topological equivalence relations

Consider $(T, \tau)$ a topological space. Now consider $\sim_1,\sim_2$ equivalence relations on T. Let's call $\sim_3= (\sim_1 \vee \sim_2$) Is it always true that that topological quotient ...
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### A problem on equivalent metrics and equivalence classes

Let $X$ be a non empty set and $\tau= \{d\mid d$ is a metric on $X\}$ Define the relation $\sim$ on $\tau$ by $d \sim d'$ iff $d$ and $d'$ are equivalent metrics on $X$. Show that $\sim$ ...
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### Why is the empty set finite?

On page 25 of Principles of Mathematical Analysis (ed. 3) by Rudin, there is the definition (excluding the irrelevant parts for this question): Definition 2.4: For any positive integer $n$, let ...
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### Equivalence relation homeomorphisms

Is said to be $X\approx Y$ ($X$ is homeomorphic to $Y$) iff exists a function $h: X \longrightarrow Y$ which is bijective and preserves open sets, this relationship is an equivalence relation on $Top$ ...
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### Do nets have subsequence?

Let $(P,\leq)$ be a directed set. Is there a cofinal and increasing function $\theta:\mathbb{N}\to P$. if $P$ has any maximal elements it can be proved easily. So I suppose $(P,\leq)$ none of it's ...
Let $X,Y$ be topological spaces. A function $\phi:X\to Y$ is continuous iff for any open subset $A\subseteq Y,$ the preimage $\phi^{-1}(A)$ is open in $X.$ We could similarly define a relation ...
How to describe all continuous maps from $T'$ to $T$, where $T=\mathbb{R}$ with natural topology (base given by the intervals $(a,b)$ ), and $T'=\mathbb{R}$ with the topology with basis given ...