Mathematics Stack Exchange
Mathematics Stack Exchange

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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Is $\mathbb Q \subset \mathbb R$ the subspace topology, discrete?, and questions about showing discreteness

I'm looking for subspace topologies of $\mathbb R$ that are discrete. In order to show that a subspace $U$ of $\mathbb R$ is discrete, I'm trying to show that the singleton set in $U$ is open. Since arbitrary unions of open sets are open, this way…
user346936
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Let $S\subset [0,1]^2$ be uncountable. Is there a line which contains infinitely many points of $S$?

If not, how many points can be guaranteed? Also, I'm not sure about my tag. This is a pretty general question. I figured General Topology is close. EDIT: Someone paraphrased this nicely. "Let $S\subset [0,1]^2$ be uncountable. Is there a line which…
user402313
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Question about the concept of connected set

From the definition, $Y$ is disconnected if $Y=A\cup B,A,B\ne\emptyset A\cap B=\emptyset$ and $A,B$ are both open. So for proving $\mathbb{Q}$ is not connected, normally the textbook would suggest…
Mathematics
  • 4,521
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Problem with the definition of a discrete topology

In wikipedia I've read that the discrete topology on X is defined by letting every subset of X be open (and hence also closed), and X is a discrete topological space if it is equipped with its discrete topology "hence also closed"? I couldn't get…
user346936
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2 answers

A quotient of a locally connected space is locally connected

Let $q : X \to Y$ be a quotient map, $X$ is a locally connected space. Show that $Y$ is also locally connected. I will be thankful if some one could present a proof of this theorem, because I couldn't find one. Thank you very much.
D. Hershko
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A cocountable topology is not first countable

I'm trying to understand this example: Let $X$ be any uncountable set and declare $U\subset X$ to be open if $X\setminus U$ is countable. Then $X$ is not first countable. Here is my reasoning: Suppose for a contradiction every $x\in X$ has a…
Sid Caroline
  • 3,729
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Is a continuous mapping which is also open (closed) must be closed(open)?

As the topic, Is a continuous mapping which is also open (closed) must be closed (open)?
Mathematics
  • 4,521
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If $A\subset\mathbb{R^2}$ is countable, is $\mathbb{R^2}\setminus A$ path connected?

Possible Duplicate: Arcwise connected part of $\mathbb R^2$ As the topic,if $A\subset\mathbb{R^2}$ is countable, does $\mathbb{R^2}\setminus A$ path connected??? I know the answer is it is path connected but not sure how to prove it.
Mathematics
  • 4,521
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Is there a set-theoretic definition of Projective Space?

I posted this on mathOverflow previously which was the wrong place to post it and I was asked to try this forum instead. Can anyone explain this in simple terms: I met projective space via a recent class on perspective drawing, believe it or not,…
mathmoggy
  • 53
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Third axiom proof

I feel stuck proving the third axiom for the topology. I proved the first two. The intersection does not seem obvious to me, and I have spent some time trying to prove it but have ran out of luck, and would really appreciate some help. Prove Theorem…
Stiven G
  • 535
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Branch of Topology

I have just started to study Topology from M. A. Armstrong's "Basic Topology". Now, I want to know which branch of Topology I can study with this book ? Or is it too early for me to worry about what I like ? If not, which book is good for the next…
Emre Yılmaz
  • 327
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Is every $\mathbb{I}$-compact topological space compact?

Let $\mathbb{I}$ denote the unit interval. Call a topological space $X$ $\mathbb{I}$-compact iff every continuous function $f : X \rightarrow \mathbb{I}$ attains its maximum at some point in $X$. Question. Is every $\mathbb{I}$-compact topological…
goblin GONE
  • 67,744
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Is every countable regular space zero-dimensional?

Question is in the title. Zero-dimensional means "has a basis of clopen sets". Hausdorff is not enough to guarantee a countable space has dimension zero (in fact, a countable Hausdorff space can be connected). Is regular enough? Note 1: I assume…
Forever Mozart
  • 9,294
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Topological space with countable open cover $\{U_\alpha\} $ with each $U_\alpha$ second-countable, is second countable

I'm trying to prove the following statement: Let $X$ be a topological space with a countable open cover $C= \{U_\alpha : \alpha \in A\}$. If each $U_\alpha$ is second-countable then $X$ is second-countable. However I'm stuck on how to proceed, so…
peter19
  • 335
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Topology generated by norm and Topology generated by the norm map

I have a problem stated, "Let $X$ be a normed space. Prove that the topology generated by norm is exactly the coarsest topology on $X$ s.t. the norm and all translations are continuous." Here's what I've done so far, it seems easy to prove that the…
T C
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