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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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Why does the product topology allow proper subsets for only finitely many elements?

Consider Theorem 19.1 from Munkres' topology: The box topology on $\prod X_\alpha$ has as basis all sets of the form $\prod U_\alpha$, where $U_\alpha$ is open in $X_\alpha$ for each $\alpha$. The product topology on $\prod X_\alpha$ has as basis…
Andrew Thompson
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Uniqueness of a continuous extension of a continuous map from a set to its closure

Suppose $f$ is a continuous map from a space $A$ to a Hausdorff space Y. Then I know that $f$ can be extended uniquely to a continuous map from closure of $A$ to Y. What is a counterexample to the fact that this need not be true if Y is not…
adrija
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Open and closed subset of $\mathbb{R}$

How to show that the only subsets of $\mathbb{R}$ which are simultaneously closed as well as open are $\emptyset$ and $\mathbb{R}$ itself. Can someone tell me how to go to the proof of it? I have tried a bit about it but no luck. My attempt…
shadow10
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Understanding a problem in Munkres

This problem is from Chapter 2, Section 16, number 5 in Munkres' Topology. This is not a homework problem, but I'm trying to complete all problems from the sections covered in class. Let $X$ and $X'$ denote a single set in the topologies $\mathcal…
dannum
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Find the closure of $\mathbb{R}^{\infty}$ in $\mathbb{R}^{\omega}$ under the box topology

Find the closure of $\mathbb{R}^{\infty}$ in $\mathbb{R}^{\omega}$ under the box topology. Note: $\mathbb{R^{\infty}}$ is the set of all sequences $(t_1,t_2,\dots)$ such that $t_i\neq0$ for only finitely many values of $i$, and…
FNH
  • 9,130
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Is the constant map a continuous function?

I've been set a question in an assignment which reads: "Check whether the following functions are continuous or open. Check whether they are a homeomorphism. $\dots$ $b)$ the constant map $f:X \rightarrow Y$ defined by $f(x)=y_0$ for some $y_0 \in…
Lammey
  • 399
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Determine whether Set is closed,whether it is open, whether it is bounded and whether it is compact

Question: Given the $$f(x,y)=y-\displaystyle\frac{1}{x^2}$$ consider the set $S = \{(x,y) \in D: x > 0,\ f(x,y) > 0\}$ where $D$ is the domain of $f$. Sketch the set $S$ in the plane. Determine whether $S$ is closed,whether it is open, whether it is…
erin
  • 125
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Conditions leading to paracompactness

If $A$ is a Hausdorff space such that $A = \bigcup\limits_{i = 1}^\infty {{K_i}} $ where $K_i$ are its compact subsets, is $A$ a paracompact space? If not, what additional conditions should we add? (e.g. locally compact)
Summer
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Algebraic condition for a twist in 2D ball or a hypersphere?

We define twisted ball here as a ball where only one point (the twist) separates the ball sides. Simple implicit presentation for the 2D Ball is $x^2+y^2=r^2$. I am trying to find a general condition when the 2D ball has a crossing like the last…
hhh
  • 5,469
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Does $A$ homeomorphic to $B$ imply $f^{-1}(A)$ is homeomorphic to $f^{-1}(B)$?

Let $X$ and $Y$ be topological spaces, and $f:X\to Y$ a continuous map. Is the following true: If $A$ and $B$ are two homeomorphic subspaces of $Y,$ then $f^{-1}(A)$ and $f^{-1}(B)$ are homeomorphic subspaces of $X$.
palio
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Connectedness of a linear continuum [Sect-24.1, Munkres]

In Munkres' Topology p.153, we have a proof like this Proof. $\ \ $ Recall that a subspace $Y$ of $L$ is said to be convex if for every pair of points $a,b$ of $Y$ with $a
Toad Jiang
  • 977
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Sufficient condition for a subset to be compact in a Hausdorff topological space

I know that if $X$ is a Hausdorff topological space and $A$ is compact in $X$, then $A$ is closed in $X$. My question is that if $A$ is a closed set in $X$ (where $X$ is Hausdorff), what extra condition is needed to ensure that $A$ is compact in…
user20694
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Are $\mathbb{R_k}$ and $\mathbb{R_L}$ really non-comparable topologies?

First of all, we notice that: $$(a,b)=\bigcup_{a
FNH
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