5
votes
2answers
159 views

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 $g$ ...
4
votes
0answers
47 views

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. ...
0
votes
0answers
18 views

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 ...
3
votes
1answer
58 views

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 ...
4
votes
3answers
93 views

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 ...
2
votes
1answer
157 views

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 $ ...
5
votes
3answers
779 views

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 ...
0
votes
1answer
76 views

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 $Top$ ...
0
votes
1answer
51 views

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 ...
11
votes
1answer
255 views

Can we extend the definition of a continuous function to binary relations?

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 ...
2
votes
0answers
146 views

How to describe all continuous maps from T to T'?

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 ...