---
title: 18.901 problem set 05
description: attempts and solutions for 18.901 problem set 05
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Problem set 5 (optional). The purpose of this problem set is to help you make up for any one of the required problem sets that you “bombed” (i.e., that you received a grade lower than 70 on). Here’s the procedure:

## 1. Properties of spaces and operations: fill in the table

> \[!question\] problem 1
>
> Select one or more rows of the following diagram. Fill in each square of that row with $+$ (for “yes”) or $-$ (for “no”). For each row filled in correctly, I will add 6 points to the score of whichever of your problem sets has the lowest score (up to a maximum of 70 points for that problem set). The same rules as usual — no collaboration, please. Entries may be submitted before or at the final exam. Decisions of the judges will be final. No box tops needed to enter!
>
> For the first six rows of the diagram, the question is this: “Is the given property satisfied by the given space?” For the last three rows, it is: “Is the given property preserved by the given operation?”
>
> |                                          | connected | path connected | compact | locally compact Hausdorff | Hausdorff | Regular | Normal | First-countable | Second-countable | Lindelöf | Has countable dense subset | Locally metrizable | Metrizable | Completely regular |
> | ---------------------------------------- | --------- | -------------- | ------- | ------------------------- | --------- | ------- | ------ | --------------- | ---------------- | -------- | -------------------------- | ------------------ | ---------- | ------------------ |
> | $S_\Omega$                               |           |                |         |                           |           |         |        |                 |                  |          |                            |                    |            |                    |
> | The ordered square                       |           |                |         |                           |           |         |        |                 |                  |          |                            |                    |            |                    |
> | $\mathbb{R}_\ell \times \mathbb{R}_\ell$ |           |                |         |                           |           |         |        |                 |                  |          |                            |                    |            |                    |
> | $\mathbb{R}^\omega$ in uniform topology  |           |                |         |                           |           |         |        |                 |                  |          |                            |                    |            |                    |
> | $\mathbb{R}^I$ in product topology       |           |                |         |                           |           |         |        |                 |                  |          |                            |                    |            |                    |
> | An arbitrary metric space                |           |                |         |                           |           |         |        |                 |                  |          |                            |                    |            |                    |
> | Taking countable products                |           |                |         |                           |           |         |        |                 |                  |          |                            |                    |            |                    |
> | Taking a closed subspace                 |           |                |         |                           |           |         |        |                 |                  |          |                            |                    |            |                    |
> | Taking an open subspace                  |           |                |         |                           |           |         |        |                 |                  |          |                            |                    |            |                    |

^problem-1

### attempt