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Questions tagged [general-topology]

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; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

57719 questions
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Are the only sets in $\mathbb{R^1}$ which are both open and closed $\mathbb{R^1}$ and $\emptyset$?

As the topic, how to prove that the only set in $\mathbb{R^1}$ which are both open and close are the $\mathbb{R^1}$ and $\emptyset$. I tried to prove by contradiction, but i can't really show that the assumption implies the contrary.
abc
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homeomorphism question relating to the topological 3-sphere

I have a question concerning an exercises from a text call Topology and Groupoid authored by Ronald Brown The question is as follows: Let $E^2 = \{(x, y) \in \mathbb R^2 : x^2 + y^2 \leq 1\}$. The space $S^1 \times E^2$ is called the solid…
Seth
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Must a space homeomorphic to $\Bbb R\setminus \Bbb Q$ have a countable complement?

There is a problem from a list suggested practice problems that I am having issues with. It says: Suppose that $X$ is a subspace of the real line $\mathbb{R}$ which is homeomorphic to the space of irrational numbers. Is the complement of $X$ in…
Maria
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Continuous self-maps on $\mathbb{Q}$

1) Are $\mathbb{Q}$ and $\mathbb{Q}\setminus\{0\}$ homeomorphic? 2) If $S\subseteq \mathbb{Q}$ is a non-empty subset, is there a continuous surjection $f:\mathbb{Q}\to S$?
Dominic van der Zypen
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$\mathbb R^2$ is not homeomorphic to $\mathbb R^3$.

I was reading Munkres for Topology. In that, it is mentioned that $\mathbb R$ is not homeomorphic to $\mathbb R^2$ as deleting a point from both makes the first one disconnected while the latter one still remains connected. Can't we say on the same…
User
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Every open set in $\mathbb{R}$ is the union of an at most countable collection of disjoint segments

Let $E$ be an open set in $\mathbb{R}$. Fix $x\in E$. I have proved that statement is true when $\{y\in \mathbb{R}|(x,y)\subset E\}$ is bounded above and $\{z\in \mathbb{R}|(z,x)\subset E\}$ is bounded below. If at least one of those above are not…
Katlus
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What differences would it make to live in $T^3$, in $S^2 \times S^1$, in $\mathbb{R}P^3$, or in $S^3$?

I am trying to conceptually understand what distinguishes these three spaces. I understand everything in terms of the fundamental domain of these spaces, i.e. the 3-cube. For $T^3$, if I set out in any one of 3 mutually perpendicular directions, I'd…
Johnver
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Perfect set in $\mathbb{R}$ which contains no rational number

Possible Duplicate: Perfect set without rationals Does there exist a nonempty perfect set in $\mathbb{R}$ which contains no rational number? This problem is on p.44 PMA - Rudin I found a proof of this on google but the proof is not 'suitable' for…
Katlus
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Connectedness of sets in the plane with rational coordinates and at least one irrational

Can someone please let me know if my solution is correct: Define: 1) Let $A = \{x \in \mathbb{R}^{2}: \text{all coordinates of x are rational} \}$. Show that $\mathbb{R}^{2} \setminus A$ is connected. My answer: just note that $A = \mathbb{Q} \times…
undergrad
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Is this statement true: a set is open if every point has a closed ball contained inside of the set

Is this statement true: Set $S$ (on a metric space) is open if $\forall x \in S$, $\exists \delta > 0,$ s.t. $\thinspace \overline B_\delta(x) \subset X$ I am a little bit thrown off by the closed ball instead of open ball definition of open set.…
Shamisen Expert
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Let $A$ be an open set of $\mathbb{R}$ and $B$ any set, under what coniditions of $B$, $AB$ is open?

I don't really know how to establish the conditions so $AB$ can be open. The problem says: Let $A$ be an open set in $\Bbb R$ and $B$ any other set. Define: $$AB = \{xy\in\mathbb{R}\,\colon x\in A\text{ and }y\in B\}$$ Is $AB$ open? I believe is…
Lore Rodriguez
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How should I think of an open vs. closed set?

I've been studying introductory topology for a little bit now. I came across this video which explains open sets in a way I have never thought of. Even though the video is pretty elementary, I didn't know that open sets worked in this sense. If I…
Rellek
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Sum of compact sets

Let $A,B$ two non-empty compact subsets of a normed space X. How can we prove that the set $S=A+B=\{a+b : a \in A, b \in B\}$ is compact? Here's my reasoning: Let $\Omega = \{\Omega_1, \Omega_2,…\}$ be an open cover of $S$. $\Omega$ induces two…
Manlio
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Do there exist space-filling curves that fill the whole plane? If so, can they be visualized?

I know that there are space-filling curves from $[0, 1]$ to the unit square, and this question addresses curves transforming the real line into the entire plane. But what about transforming the unit interval into the whole plane? And if such curves…
kuzzooroo
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The closed set in the product topology

We know that for the product topology $X\times Y$, the open sets are generated by $U\times V$,where $U,V$ are open in $X,Y$ respectively. I am considering the closed sets in $X\times Y$, are they generated by the closed sets in $X$ and $Y$?
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