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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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Prove that a subset of meager set is also meager

I have to prove that a subset $A \subset F$ of meager set $F$ is also a meager set. From the definition of meager set $F = \bigcup_{n=1}^{\infty}F_n$, where $F_n$ is a nowhere dense set. As I know we have to show that $A$ has also similiar form, but…
janusz
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Is a $\sigma$ - compact Hausdorff space normal?

Are $\sigma$ -compact Hausdorff spaces normal?
Nicolas Bourbaki
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7
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subbasis for a topology Munkres

I have a question about the following definition: A subbasis $S$ for a topology on a set $X$ is a collection of subsets of $X$ whose union equals $X$. The topology generated by the subbasis $S$ is defined to be the collection $T$ of all unions of…
Vien Nguyen
  • 1,564
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Is there an interval in the intersection of compact subsets under such a condition?

Suppose $A$ be an index set, and $A$ is uncountable. $\mathcal{F} = \{F_a \}_{a \in A}$ is a family of compact subsets in $\mathbb{R},$ that is, $F_a$ is a compact subset in $\mathbb{R}$ for any $a \in A$. Suppose for any countable subset $I…
Haipeng Chen
  • 278
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Infinitely many discontinuities for a bijective function from $[0,1)$ to $(0,1)$

Show that any bijection from $[0,1)$ to $(0,1)$ has infinitely many discontinuities. I have thought about this question but I have no any idea. Any idea is valuable for me, thanks.
user365
  • 137
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continuous constant function general topology

I've read that if $X$ and $Y$ are topological spaces, and if the constant function $f: X \rightarrow Y$ mapping all of $X$ into the single point $y_{0}$ of $Y$, then $f$ is continuous. (Munkres page 107) Munkres gives the following brief proof:…
Vien Nguyen
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2 answers

How is that $S^1$ is not contractible?

It is stated in Wikipedia (and other pages too) that the spheres $S^n$ are all not contractible. Take $n=1$. Would anyone explain to me why $$S^1\times [0,1]\to S^1$$$$(e^{2\pi i t},s)\mapsto e^{2\pi i ts}$$is not an homotopy between the identity…
Talexius
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Topology of a circle

I'm currently trying to learn topology by myself, before I started I knew topology had something to do with shaped stretched but not torn or glued. Now that I have started to learn I see talk about open and closed sets, neighborhoods, limits and…
Adi Ostrov
  • 261
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Limit point and interior point

Is any interior point also a limit point? Judging from the definition, I believe every interior point is a limit point, but I'm not sure about it. If this is wrong, could you give me a counterexample? (Since an interior point $p$ of a set $E$ has a…
Tengu
  • 1,078
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Continuous maps in topology; the definition?

I am just wondering, given the definition of continuous maps as follows, A functionn $f:X \to Y$ is continuous if for every open subset $U $ of $Y$ the preimage $f^{-1}U$ is open in $X$. I guess mathematically, this doesn't necessarily mean that…
John Trail
  • 3,279
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Connected Components for $(\Bbb R, \mathcal T_{ lower limit})$

$(\Bbb R, \mathcal T_{{ lower }{ limit}})$ is a topological space $\Bbb R$ with Lower limit topology. As I know, $(\Bbb R, \mathcal T_{{ lower }{ limit}})$ is disconnected. What are the Connected Components of $(\Bbb R, \mathcal T_{ lower limit})$…
Belive
  • 445
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It there an example of sum and product of continuous functions is not continuous?

The sum and product of two continuous functions is continuous I can prove this easily when the space is metrizable, but I don't get it when the space is non-metrizable. Is there a counterexample of this? or it is true for all topological spaces…
Jay C. Lee
  • 171
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Showing that the closure of the closure of a set is its closure

I have the feeling I'm missing something obvious, but here it goes... I'm trying to prove that for a subset $A$ of a topological space $X$, $\overline{\overline{A}}=\overline{A}$. The inclusion $\overline{\overline{A}} \subseteq \overline{A}$ I can…
Alex Petzke
  • 8,763
7
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2 answers

Uncountably many points of an uncountable set in a second countable are limit points

I'm trying to solve the following exercise from Munkres book. Can you please check it? Let $X$ be a second countable space and let $A$ be an uncountable subset of $X$. Show that an uncountable number of points of $A$ are limit points of…
student
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Lower limit topology is normal

How do I prove that the space of real numbers, under the lower limit topology, is a normal space. I could prove very easily that it is regular, by using an argument of basic sets, but I haven't been able to generalise that argument.
MathManiac
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