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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 rationals minus a point homeomorphic to the rationals?

A while ago I was dreaming up point-set topology exam questions, and this one came to mind: Is $\mathbb Q\setminus \{0\}$ homeomorphic to $\mathbb Q$? (Where both sets have the subspace topology induced from the standard topology on $\mathbb…
Cheerful Parsnip
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3 answers

Is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$

I need a hint. The problem is: is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$ I'm pretty sure that there aren't any, but so far I couldn't find the proof. My best idea so far is to consider $f' = f|_{\mathbb{R}-\{*\}}:…
Aleksei Averchenko
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Why is Completeness not a Topological Property?

I am trying to answer the question: Show why completeness is not a topological property. My answer: $\mathbb{R}$ and the set $(0,1)$ are homeomorphic, but $\mathbb{R}$ is complete while $(0,1)$ is not. My question to you all: Does this answer the…
Kara
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7 answers

Are Singleton sets in $\mathbb{R}$ both closed and open?

I think singleton sets $\{x\}$ where $x$ is a member of $\mathbb{R}$ are both open and closed. Singleton sets are open because $\{x\}$ is a subset of itself. There are no points in the neighborhood of $x$. I want to know singleton sets are closed or…
Vinod
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34
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3 answers

Connected sum of projective plane $\cong$ Klein bottle

How can I see that the connected sum $\mathbb{P}^2 \# \mathbb{P}^2$ of the projective plane is homeomorphic to the Klein bottle? I'm not necessarily looking for an explicit homeomorphism, just an intuitive argument of why this is the case. Can we…
iwriteonbananas
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6 answers

An example of neither open nor closed set

I need a very simple example of a set of real numbers (if there is any) that is neither closed nor open, along with an explanation of why it is so.
Monkey D. Luffy
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33
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8 answers

Definition of neighborhood and open set in topology

I am a Physics undergrad, and just started studying Topology. How do you define neighborhood and open set in Topology.Wikipedia gives a circular definition. An open set is defined as follows. In topology, a set is called an open set if it is a…
user23238
32
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1 answer

Is there a Hausdorff space $X$ separating points of any Hausdorff space?

There is a nice property of normal spaces, namely, closed disjoint subsets can be separated by continuous functions into $\mathbb R$. Then you ask yourself, what about Hausdorff spaces?, are all Hausdorff spaces completely Hausdorff? Well, if we…
Camilo Arosemena-Serrato
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3 answers

Continuous Functions from $\mathbb{R}$ to $\mathbb{Q}$

The following is not a homework problem. I am doing it for self study. Prove that any continuous function from $\mathbb{R}$ to $\mathbb{Q}$ is constant. Here is my proof: Let $f:\mathbb{R}\rightarrow \mathbb{Q}$ be such a function. ? We first…
Holdsworth88
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Arbitrary intersection of closed, connected subsets of a compact space connected?

Let $(B_i)_{i\in I}$ be an indexed family of closed, connected sets in a compact space X. Suppose $I$ is ordered, sucht that $i < j \implies B_i \supset B_j$. Is $B = \bigcap_i B_i$ necessarily connected? I can prove it, if I assume $X$ to be…
Sam
  • 15,908
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6 answers

Is every open set the interior of a closed set?

I am wondering if this is generally true for any topology. I think there might be counter examples, but I am having trouble generating them.
user160110
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The graph of xy = 1 is connected or not

The graph of $xy = 1$ in $\Bbb C^2$ is connected. True or false? I know that it is not connected in $\Bbb R^2$, but what is the case of $\Bbb C^2$?
poton
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6 answers

meaning of topology and topological space

After looking at the Wikipedia article on topological space, I still cannot grasp intuitively what topological space is. For example, if we are to define topology on real numbers, can there be many topological space models, and why is defining…
user27515
  • 895
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8 answers

What is the mathematical distinction between closed and open sets?

If you wanted me to spell out the difference between closed and open sets, the best I could do is to draw you a circle one with dotted circumference the other with continuous circumference. Or I would give you an example with a set of numbers $(1,…
Shamisen Expert
  • 8,092
30
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2 answers

In which topological spaces is every singleton set a zero set?

The title question says it all: if $X$ is a topological space, then a subset $Z$ of $X$ is a zero set if there is a continuous function $f: X \rightarrow \mathbb{R}$ with $Z = f^{-1}(0)$. Now I know the following: Every zero set is a closed…
Pete L. Clark
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